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id="toc-Historical_notation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Historical_notation"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Historical notation</span> </div> </a> <ul id="toc-Historical_notation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-First_use_of_the_term" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#First_use_of_the_term"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>First use of the term</span> </div> </a> <ul id="toc-First_use_of_the_term-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Terminology_and_notation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Terminology_and_notation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Terminology and notation</span> </div> </a> <ul id="toc-Terminology_and_notation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Interpretations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Interpretations"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Interpretations</span> </div> </a> <ul id="toc-Interpretations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Formal_definitions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Formal_definitions"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Formal definitions</span> </div> </a> <button aria-controls="toc-Formal_definitions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Formal definitions subsection</span> </button> <ul id="toc-Formal_definitions-sublist" class="vector-toc-list"> <li id="toc-Riemann_integral" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Riemann_integral"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Riemann integral</span> </div> </a> <ul id="toc-Riemann_integral-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lebesgue_integral" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lebesgue_integral"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Lebesgue integral</span> </div> </a> <ul id="toc-Lebesgue_integral-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_integrals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_integrals"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Other integrals</span> </div> </a> <ul id="toc-Other_integrals-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Properties</span> </div> </a> <button aria-controls="toc-Properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Properties subsection</span> </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Linearity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Linearity"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Linearity</span> </div> </a> <ul id="toc-Linearity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inequalities" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inequalities"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Inequalities</span> </div> </a> <ul id="toc-Inequalities-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conventions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conventions"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Conventions</span> </div> </a> <ul id="toc-Conventions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Fundamental_theorem_of_calculus" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Fundamental_theorem_of_calculus"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Fundamental theorem of calculus</span> </div> </a> <button aria-controls="toc-Fundamental_theorem_of_calculus-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Fundamental theorem of calculus subsection</span> </button> <ul id="toc-Fundamental_theorem_of_calculus-sublist" class="vector-toc-list"> <li id="toc-First_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#First_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>First theorem</span> </div> </a> <ul id="toc-First_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Second_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Second_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Second theorem</span> </div> </a> <ul id="toc-Second_theorem-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Extensions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Extensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Extensions</span> </div> </a> <button aria-controls="toc-Extensions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Extensions subsection</span> </button> <ul id="toc-Extensions-sublist" class="vector-toc-list"> <li id="toc-Improper_integrals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Improper_integrals"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Improper integrals</span> </div> </a> <ul id="toc-Improper_integrals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multiple_integration" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multiple_integration"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Multiple integration</span> </div> </a> <ul id="toc-Multiple_integration-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Line_integrals_and_surface_integrals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Line_integrals_and_surface_integrals"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>Line integrals and surface integrals</span> </div> </a> <ul id="toc-Line_integrals_and_surface_integrals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Contour_integrals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Contour_integrals"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.4</span> <span>Contour integrals</span> </div> </a> <ul id="toc-Contour_integrals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Integrals_of_differential_forms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integrals_of_differential_forms"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.5</span> <span>Integrals of differential forms</span> </div> </a> <ul id="toc-Integrals_of_differential_forms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Summations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Summations"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.6</span> <span>Summations</span> </div> </a> <ul id="toc-Summations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Functional_integrals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Functional_integrals"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.7</span> <span>Functional integrals</span> </div> </a> <ul id="toc-Functional_integrals-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Computation"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Computation</span> </div> </a> <button aria-controls="toc-Computation-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Computation subsection</span> </button> <ul id="toc-Computation-sublist" class="vector-toc-list"> <li id="toc-Analytical" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Analytical"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Analytical</span> </div> </a> <ul id="toc-Analytical-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Symbolic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Symbolic"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.2</span> <span>Symbolic</span> </div> </a> <ul id="toc-Symbolic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Numerical" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Numerical"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.3</span> <span>Numerical</span> </div> </a> <ul id="toc-Numerical-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mechanical" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mechanical"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.4</span> <span>Mechanical</span> </div> </a> <ul id="toc-Mechanical-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geometrical" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geometrical"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.5</span> <span>Geometrical</span> </div> </a> <ul id="toc-Geometrical-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Integration_by_differentiation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integration_by_differentiation"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.6</span> <span>Integration by differentiation</span> </div> </a> <ul id="toc-Integration_by_differentiation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Examples</span> </div> </a> <button aria-controls="toc-Examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Examples subsection</span> </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Using_the_fundamental_theorem_of_calculus" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Using_the_fundamental_theorem_of_calculus"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.1</span> <span>Using the fundamental theorem of calculus</span> </div> </a> <ul id="toc-Using_the_fundamental_theorem_of_calculus-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>Bibliography</span> </div> </a> <ul id="toc-Bibliography-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">15</span> <span>External links</span> </div> </a> <button aria-controls="toc-External_links-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle External links subsection</span> </button> <ul id="toc-External_links-sublist" class="vector-toc-list"> <li id="toc-Online_books" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Online_books"> <div class="vector-toc-text"> <span class="vector-toc-numb">15.1</span> <span>Online books</span> </div> </a> <ul id="toc-Online_books-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Integral</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 92 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-92" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">92 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8A%A0%E1%8C%A0%E1%88%AB%E1%89%83%E1%88%9A" title="አጠራቃሚ – Amharic" lang="am" hreflang="am" data-title="አጠራቃሚ" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D9%83%D8%A7%D9%85%D9%84" title="تكامل – Arabic" lang="ar" hreflang="ar" data-title="تكامل" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Integraci%C3%B3n" title="Integración – Aragonese" lang="an" hreflang="an" data-title="Integración" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Integraci%C3%B3n" title="Integración – Asturian" lang="ast" hreflang="ast" data-title="Integración" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C4%B0nteqral" title="İnteqral – Azerbaijani" lang="az" hreflang="az" data-title="İnteqral" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%A7%D9%86%D8%AA%D9%82%D8%B1%D8%A7%D9%84" title="انتقرال – South Azerbaijani" lang="azb" hreflang="azb" data-title="انتقرال" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AF%E0%A7%8B%E0%A6%97%E0%A6%9C%E0%A7%80%E0%A6%95%E0%A6%B0%E0%A6%A3" title="যোগজীকরণ – Bangla" lang="bn" hreflang="bn" data-title="যোগজীকরণ" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Chek-hun" title="Chek-hun – Minnan" lang="nan" hreflang="nan" data-title="Chek-hun" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%98%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB" title="Интеграл – Bashkir" lang="ba" hreflang="ba" data-title="Интеграл" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%86%D0%BD%D1%82%D1%8D%D0%B3%D1%80%D0%B0%D0%BB" title="Інтэграл – Belarusian" lang="be" hreflang="be" data-title="Інтэграл" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%98%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB" title="Интеграл – Bulgarian" lang="bg" hreflang="bg" data-title="Интеграл" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Integral" title="Integral – Bosnian" lang="bs" hreflang="bs" data-title="Integral" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://ca.wikipedia.org/wiki/Integraci%C3%B3" title="Integració – Catalan" lang="ca" hreflang="ca" data-title="Integració" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%98%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB" title="Интеграл – Chuvash" lang="cv" hreflang="cv" data-title="Интеграл" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Integr%C3%A1l" title="Integrál – Czech" lang="cs" hreflang="cs" data-title="Integrál" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Integryn" title="Integryn – Welsh" lang="cy" hreflang="cy" data-title="Integryn" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-de badge-Q70894304 mw-list-item" title=""><a href="https://de.wikipedia.org/wiki/Integral" title="Integral – German" lang="de" hreflang="de" data-title="Integral" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Integraal" title="Integraal – Estonian" lang="et" hreflang="et" data-title="Integraal" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9F%CE%BB%CE%BF%CE%BA%CE%BB%CE%AE%CF%81%CF%89%CE%BC%CE%B1" title="Ολοκλήρωμα – Greek" lang="el" hreflang="el" data-title="Ολοκλήρωμα" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Integraci%C3%B3n" title="Integración – Spanish" lang="es" hreflang="es" data-title="Integración" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Integralo" title="Integralo – Esperanto" lang="eo" hreflang="eo" data-title="Integralo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://eu.wikipedia.org/wiki/Integral" title="Integral – Basque" lang="eu" hreflang="eu" data-title="Integral" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D9%86%D8%AA%DA%AF%D8%B1%D8%A7%D9%84" title="انتگرال – Persian" lang="fa" hreflang="fa" data-title="انتگرال" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Int%C3%A9gration_(math%C3%A9matiques)" title="Intégration (mathématiques) – French" lang="fr" hreflang="fr" data-title="Intégration (mathématiques)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Integral" title="Integral – Galician" lang="gl" hreflang="gl" data-title="Integral" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%B8%E0%AA%82%E0%AA%95%E0%AA%B2%E0%AA%A8" title="સંકલન – Gujarati" lang="gu" hreflang="gu" data-title="સંકલન" data-language-autonym="ગુજરાતી" data-language-local-name="Gujarati" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-hak mw-list-item"><a href="https://hak.wikipedia.org/wiki/Chit-f%C3%BBn-ho%CC%8Dk" title="Chit-fûn-ho̍k – Hakka Chinese" lang="hak" hreflang="hak" data-title="Chit-fûn-ho̍k" data-language-autonym="客家語 / Hak-kâ-ngî" data-language-local-name="Hakka Chinese" class="interlanguage-link-target"><span>客家語 / Hak-kâ-ngî</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%81%EB%B6%84" title="적분 – Korean" lang="ko" hreflang="ko" data-title="적분" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BB%D5%B6%D5%BF%D5%A5%D5%A3%D6%80%D5%A1%D5%AC" title="Ինտեգրալ – Armenian" lang="hy" hreflang="hy" data-title="Ինտեգրալ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%AE%E0%A4%BE%E0%A4%95%E0%A4%B2%E0%A4%A8" title="समाकलन – Hindi" lang="hi" hreflang="hi" data-title="समाकलन" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Integral" title="Integral – Croatian" lang="hr" hreflang="hr" data-title="Integral" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Integralo" title="Integralo – Ido" lang="io" hreflang="io" data-title="Integralo" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Integral" title="Integral – Indonesian" lang="id" hreflang="id" data-title="Integral" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Heildun" title="Heildun – Icelandic" lang="is" hreflang="is" data-title="Heildun" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Integrale" title="Integrale – Italian" lang="it" hreflang="it" data-title="Integrale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C" title="אינטגרל – Hebrew" lang="he" hreflang="he" data-title="אינטגרל" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%85%E0%B2%A8%E0%B3%81%E0%B2%95%E0%B2%B2%E0%B2%A8%E0%B2%B6%E0%B2%BE%E0%B2%B8%E0%B3%8D%E0%B2%A4%E0%B3%8D%E0%B2%B0" title="ಅನುಕಲನಶಾಸ್ತ್ರ – Kannada" lang="kn" hreflang="kn" data-title="ಅನುಕಲನಶಾಸ್ತ್ರ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%98%E1%83%9C%E1%83%A2%E1%83%94%E1%83%92%E1%83%A0%E1%83%90%E1%83%9A%E1%83%98" title="ინტეგრალი – Georgian" lang="ka" hreflang="ka" data-title="ინტეგრალი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%98%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB" title="Интеграл – Kazakh" lang="kk" hreflang="kk" data-title="Интеграл" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Ukamilishaji_(hisabati)" title="Ukamilishaji (hisabati) – Swahili" lang="sw" hreflang="sw" data-title="Ukamilishaji (hisabati)" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Entegrasyon_(matematik)" title="Entegrasyon (matematik) – Haitian Creole" lang="ht" hreflang="ht" data-title="Entegrasyon (matematik)" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/%C3%8Entegral" title="Întegral – Kurdish" lang="ku" hreflang="ku" data-title="Întegral" data-language-autonym="Kurdî" data-language-local-name="Kurdish" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%98%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB" title="Интеграл – Kyrgyz" lang="ky" hreflang="ky" data-title="Интеграл" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Integrale" title="Integrale – Latin" lang="la" hreflang="la" data-title="Integrale" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Integr%C4%81lis" title="Integrālis – Latvian" lang="lv" hreflang="lv" data-title="Integrālis" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Integralas" title="Integralas – Lithuanian" lang="lt" hreflang="lt" data-title="Integralas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Integral" title="Integral – Lombard" lang="lmo" hreflang="lmo" data-title="Integral" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Integr%C3%A1l" title="Integrál – Hungarian" lang="hu" hreflang="hu" data-title="Integrál" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B8%E0%B4%AE%E0%B4%BE%E0%B4%95%E0%B4%B2%E0%B4%A8%E0%B4%82" title="സമാകലനം – Malayalam" lang="ml" hreflang="ml" data-title="സമാകലനം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/L-Integral" title="L-Integral – Maltese" lang="mt" hreflang="mt" data-title="L-Integral" data-language-autonym="Malti" data-language-local-name="Maltese" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B8%E0%A4%82%E0%A4%95%E0%A4%B2%E0%A4%A8" title="संकलन – Marathi" lang="mr" hreflang="mr" data-title="संकलन" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Kamiran" title="Kamiran – Malay" lang="ms" hreflang="ms" data-title="Kamiran" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%98%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB" title="Интеграл – Mongolian" lang="mn" hreflang="mn" data-title="Интеграл" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%A1%E1%80%84%E1%80%BA%E1%80%90%E1%80%AE%E1%80%82%E1%80%9B%E1%80%B1%E1%80%B8%E1%80%9B%E1%80%BE%E1%80%84%E1%80%BA%E1%80%B8" title="အင်တီဂရေးရှင်း – Burmese" lang="my" hreflang="my" data-title="အင်တီဂရေးရှင်း" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl badge-Q70894304 mw-list-item" title=""><a href="https://nl.wikipedia.org/wiki/Integraal" title="Integraal – Dutch" lang="nl" hreflang="nl" data-title="Integraal" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Integrasjon" title="Integrasjon – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Integrasjon" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Integral" title="Integral – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Integral" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Integracion" title="Integracion – Occitan" lang="oc" hreflang="oc" data-title="Integracion" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Wajjummaa" title="Wajjummaa – Oromo" lang="om" hreflang="om" data-title="Wajjummaa" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Integral" title="Integral – Uzbek" lang="uz" hreflang="uz" data-title="Integral" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%A7%D9%86%D9%B9%DB%8C%DA%AF%D8%B1%D9%84" title="انٹیگرل – Western Punjabi" lang="pnb" hreflang="pnb" data-title="انٹیگرل" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%A2%E1%9E%B6%E1%9F%86%E1%9E%84%E1%9E%8F%E1%9F%81%E1%9E%80%E1%9F%92%E1%9E%9A%E1%9E%B6%E1%9E%9B" title="អាំងតេក្រាល – Khmer" lang="km" hreflang="km" data-title="អាំងតេក្រាល" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Ca%C5%82ka" title="Całka – Polish" lang="pl" hreflang="pl" data-title="Całka" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Integral" title="Integral – Portuguese" lang="pt" hreflang="pt" data-title="Integral" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-kaa mw-list-item"><a href="https://kaa.wikipedia.org/wiki/Integral" title="Integral – Kara-Kalpak" lang="kaa" hreflang="kaa" data-title="Integral" data-language-autonym="Qaraqalpaqsha" data-language-local-name="Kara-Kalpak" class="interlanguage-link-target"><span>Qaraqalpaqsha</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Integral%C4%83" title="Integrală – Romanian" lang="ro" hreflang="ro" data-title="Integrală" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%98%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB" title="Интеграл – Russian" lang="ru" hreflang="ru" data-title="Интеграл" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Integral" title="Integral – Scots" lang="sco" hreflang="sco" data-title="Integral" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Integrali" title="Integrali – Albanian" lang="sq" hreflang="sq" data-title="Integrali" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Intiggrali" title="Intiggrali – Sicilian" lang="scn" hreflang="scn" data-title="Intiggrali" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Integral" title="Integral – Simple English" lang="en-simple" hreflang="en-simple" data-title="Integral" data-language-autonym="Simple 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class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Integral_calculus&redirect=no" class="mw-redirect" title="Integral calculus">Integral calculus</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Operation in mathematical calculus</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about the concept of definite integrals in calculus. For the indefinite integral, see <a href="/wiki/Antiderivative" title="Antiderivative">antiderivative</a>. For the set of numbers, see <a href="/wiki/Integer" title="Integer">integer</a>. For other uses, see <a href="/wiki/Integral_(disambiguation)" class="mw-disambig" title="Integral (disambiguation)">Integral (disambiguation)</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Area under the curve" redirects here. For the pharmacology integral, see <a href="/wiki/Area_under_the_curve_(pharmacokinetics)" title="Area under the curve (pharmacokinetics)">Area under the curve (pharmacokinetics)</a>. For the statistics concept, see <a href="/wiki/Receiver_operating_characteristic#Area_under_the_curve" title="Receiver operating characteristic">Receiver operating characteristic § Area under the curve</a>.</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Integral_example.svg" class="mw-file-description"><img alt="Definite integral example" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Integral_example.svg/300px-Integral_example.svg.png" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Integral_example.svg/450px-Integral_example.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Integral_example.svg/600px-Integral_example.svg.png 2x" data-file-width="420" data-file-height="420" /></a><figcaption>A definite integral of a function can be represented as the <a href="/wiki/Signed_area" title="Signed area">signed area</a> of the region bounded by its graph and the horizontal axis; in the above graph as an example, the integral of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> is the yellow (−) area subtracted from the blue (+) area</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output 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href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><td class="sidebar-pretitle">Part of a series of articles about</td></tr><tr><th class="sidebar-title-with-pretitle" style="padding-bottom:0.25em;"><a href="/wiki/Calculus" title="Calculus">Calculus</a></th></tr><tr><td class="sidebar-image"><big><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17d063dc86a53a2efb1fe86f4a5d47d498652766" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.228ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}"></span></big></td></tr><tr><td class="sidebar-above" style="padding:0.15em 0.25em 0.3em;font-weight:normal;"> <ul><li><a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">Fundamental theorem</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Limit_of_a_function" title="Limit of a function">Limits</a></li> <li><a href="/wiki/Continuous_function" title="Continuous function">Continuity</a></li></ul> </div><div class="hlist"> <ul><li><a href="/wiki/Rolle%27s_theorem" title="Rolle's theorem">Rolle's theorem</a></li> <li><a href="/wiki/Mean_value_theorem" title="Mean value theorem">Mean value theorem</a></li> <li><a href="/wiki/Inverse_function_theorem" title="Inverse function theorem">Inverse function theorem</a></li></ul> </div></td></tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base);display:block;margin-top:0.65em;"><span style="font-size:120%"><a href="/wiki/Differential_calculus" title="Differential calculus">Differential</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading"> Definitions</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Derivative" title="Derivative">Derivative</a> (<a href="/wiki/Generalizations_of_the_derivative" title="Generalizations of the derivative">generalizations</a>)</li> <li><a href="/wiki/Differential_(mathematics)" title="Differential (mathematics)">Differential</a> <ul><li><a href="/wiki/Differential_(infinitesimal)" class="mw-redirect" title="Differential (infinitesimal)">infinitesimal</a></li> <li><a href="/wiki/Differential_of_a_function" title="Differential of a function">of a function</a></li> <li><a href="/wiki/Differential_of_a_function#Differentials_in_several_variables" title="Differential of a function">total</a></li></ul></li></ul></td> </tr><tr><th class="sidebar-heading"> Concepts</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Notation_for_differentiation" title="Notation for differentiation">Differentiation notation</a></li> <li><a href="/wiki/Second_derivative" title="Second derivative">Second derivative</a></li> <li><a href="/wiki/Implicit_function" title="Implicit function">Implicit differentiation</a></li> <li><a href="/wiki/Logarithmic_differentiation" title="Logarithmic differentiation">Logarithmic differentiation</a></li> <li><a href="/wiki/Related_rates" title="Related rates">Related rates</a></li> <li><a href="/wiki/Taylor%27s_theorem" title="Taylor's theorem">Taylor's theorem</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Differentiation_rules" title="Differentiation rules">Rules and identities</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Sum_rule_in_differentiation" class="mw-redirect" title="Sum rule in differentiation">Sum</a></li> <li><a href="/wiki/Product_rule" title="Product rule">Product</a></li> <li><a href="/wiki/Chain_rule" title="Chain rule">Chain</a></li> <li><a href="/wiki/Power_rule" title="Power rule">Power</a></li> <li><a href="/wiki/Quotient_rule" title="Quotient rule">Quotient</a></li> <li><a href="/wiki/L%27H%C3%B4pital%27s_rule" title="L'Hôpital's rule">L'Hôpital's rule</a></li> <li><a href="/wiki/Inverse_function_rule" title="Inverse function rule">Inverse</a></li> <li><a href="/wiki/General_Leibniz_rule" title="General Leibniz rule">General Leibniz</a></li> <li><a href="/wiki/Fa%C3%A0_di_Bruno%27s_formula" title="Faà di Bruno's formula">Faà di Bruno's formula</a></li> <li><a href="/wiki/Reynolds_transport_theorem" title="Reynolds transport theorem">Reynolds</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%"><a class="mw-selflink selflink">Integral</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Lists_of_integrals" title="Lists of integrals">Lists of integrals</a></li> <li><a href="/wiki/Integral_transform" title="Integral transform">Integral transform</a></li> <li><a href="/wiki/Leibniz_integral_rule" title="Leibniz integral rule">Leibniz integral rule</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Definitions</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Antiderivative" title="Antiderivative">Antiderivative</a></li> <li><a class="mw-selflink selflink">Integral</a> (<a href="/wiki/Improper_integral" title="Improper integral">improper</a>)</li> <li><a href="/wiki/Riemann_integral" title="Riemann integral">Riemann integral</a></li> <li><a href="/wiki/Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue integration</a></li> <li><a href="/wiki/Contour_integration" title="Contour integration">Contour integration</a></li> <li><a href="/wiki/Integral_of_inverse_functions" title="Integral of inverse functions">Integral of inverse functions</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Integration by</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Integration_by_parts" title="Integration by parts">Parts</a></li> <li><a href="/wiki/Disc_integration" title="Disc integration">Discs</a></li> <li><a href="/wiki/Shell_integration" title="Shell integration">Cylindrical shells</a></li> <li><a href="/wiki/Integration_by_substitution" title="Integration by substitution">Substitution</a> (<a href="/wiki/Trigonometric_substitution" title="Trigonometric substitution">trigonometric</a>, <a href="/wiki/Tangent_half-angle_substitution" title="Tangent half-angle substitution">tangent half-angle</a>, <a href="/wiki/Euler_substitution" title="Euler substitution">Euler</a>)</li> <li><a href="/wiki/Integration_using_Euler%27s_formula" title="Integration using Euler's formula">Euler's formula</a></li> <li><a href="/wiki/Partial_fractions_in_integration" class="mw-redirect" title="Partial fractions in integration">Partial fractions</a> (<a href="/wiki/Heaviside_cover-up_method" title="Heaviside cover-up method">Heaviside's method</a>)</li> <li><a href="/wiki/Order_of_integration_(calculus)" title="Order of integration (calculus)">Changing order</a></li> <li><a href="/wiki/Integration_by_reduction_formulae" title="Integration by reduction formulae">Reduction formulae</a></li> <li><a href="/wiki/Leibniz_integral_rule#Evaluating_definite_integrals" title="Leibniz integral rule">Differentiating under the integral sign</a></li> <li><a href="/wiki/Risch_algorithm" title="Risch algorithm">Risch algorithm</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%"><a href="/wiki/Series_(mathematics)" title="Series (mathematics)">Series</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Geometric_series" title="Geometric series">Geometric</a> (<a href="/wiki/Arithmetico%E2%80%93geometric_sequence" class="mw-redirect" title="Arithmetico–geometric sequence">arithmetico-geometric</a>)</li> <li><a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">Harmonic</a></li> <li><a href="/wiki/Alternating_series" title="Alternating series">Alternating</a></li> <li><a href="/wiki/Power_series" title="Power series">Power</a></li> <li><a href="/wiki/Binomial_series" title="Binomial series">Binomial</a></li> <li><a href="/wiki/Taylor_series" title="Taylor series">Taylor</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Convergence_tests" title="Convergence tests">Convergence tests</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Term_test" class="mw-redirect" title="Term test">Summand limit (term test)</a></li> <li><a href="/wiki/Ratio_test" title="Ratio test">Ratio</a></li> <li><a href="/wiki/Root_test" title="Root test">Root</a></li> <li><a href="/wiki/Integral_test_for_convergence" title="Integral test for convergence">Integral</a></li> <li><a href="/wiki/Direct_comparison_test" title="Direct comparison test">Direct comparison</a></li> <li><br /><a href="/wiki/Limit_comparison_test" title="Limit comparison test">Limit comparison</a></li> <li><a href="/wiki/Alternating_series_test" title="Alternating series test">Alternating series</a></li> <li><a href="/wiki/Cauchy_condensation_test" title="Cauchy condensation test">Cauchy condensation</a></li> <li><a href="/wiki/Dirichlet%27s_test" title="Dirichlet's test">Dirichlet</a></li> <li><a href="/wiki/Abel%27s_test" title="Abel's test">Abel</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%"><a href="/wiki/Vector_calculus" title="Vector calculus">Vector</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Gradient" title="Gradient">Gradient</a></li> <li><a href="/wiki/Divergence" title="Divergence">Divergence</a></li> <li><a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)">Curl</a></li> <li><a href="/wiki/Laplace_operator" title="Laplace operator">Laplacian</a></li> <li><a href="/wiki/Directional_derivative" title="Directional derivative">Directional derivative</a></li> <li><a href="/wiki/Vector_calculus_identities" title="Vector calculus identities">Identities</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Theorems</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Gradient_theorem" title="Gradient theorem">Gradient</a></li> <li><a href="/wiki/Green%27s_theorem" title="Green's theorem">Green's</a></li> <li><a href="/wiki/Stokes%27_theorem" title="Stokes' theorem">Stokes'</a></li> <li><a href="/wiki/Divergence_theorem" title="Divergence theorem">Divergence</a></li> <li><a href="/wiki/Generalized_Stokes_theorem" title="Generalized Stokes theorem">generalized Stokes</a></li> <li><a href="/wiki/Helmholtz_decomposition" title="Helmholtz decomposition">Helmholtz decomposition</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%"><a href="/wiki/Multivariable_calculus" title="Multivariable calculus">Multivariable</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading"> Formalisms</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Matrix_calculus" title="Matrix calculus">Matrix</a></li> <li><a href="/wiki/Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor</a></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior</a></li> <li><a href="/wiki/Geometric_calculus" title="Geometric calculus">Geometric</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Definitions</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Partial_derivative" title="Partial derivative">Partial derivative</a></li> <li><a href="/wiki/Multiple_integral" title="Multiple integral">Multiple integral</a></li> <li><a href="/wiki/Line_integral" title="Line integral">Line integral</a></li> <li><a href="/wiki/Surface_integral" title="Surface integral">Surface integral</a></li> <li><a href="/wiki/Volume_integral" title="Volume integral">Volume integral</a></li> <li><a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a></li> <li><a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessian</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%">Advanced</span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Calculus_on_Euclidean_space" title="Calculus on Euclidean space">Calculus on Euclidean space</a></li> <li><a href="/wiki/Generalized_function" title="Generalized function">Generalized functions</a></li> <li><a href="/wiki/Limit_of_distributions" title="Limit of distributions">Limit of distributions</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%">Specialized</span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Fractional_calculus" title="Fractional calculus">Fractional</a></li> <li><a href="/wiki/Malliavin_calculus" title="Malliavin calculus">Malliavin</a></li> <li><a href="/wiki/Stochastic_calculus" title="Stochastic calculus">Stochastic</a></li> <li><a href="/wiki/Calculus_of_variations" title="Calculus of variations">Variations</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%">Miscellanea</span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Precalculus" title="Precalculus">Precalculus</a></li> <li><a href="/wiki/History_of_calculus" title="History of calculus">History</a></li> <li><a href="/wiki/Glossary_of_calculus" title="Glossary of calculus">Glossary</a></li> <li><a href="/wiki/List_of_calculus_topics" title="List of calculus topics">List of topics</a></li> <li><a href="/wiki/Integration_Bee" title="Integration Bee">Integration Bee</a></li> <li><a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Mathematical analysis</a></li> <li><a href="/wiki/Nonstandard_analysis" title="Nonstandard analysis">Nonstandard analysis</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Calculus" title="Template:Calculus"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Calculus" title="Template talk:Calculus"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Calculus" title="Special:EditPage/Template:Calculus"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, an <b>integral</b> is the continuous analog of a <a href="/wiki/Summation" title="Summation">sum</a>, which is used to calculate <a href="/wiki/Area" title="Area">areas</a>, <a href="/wiki/Volume" title="Volume">volumes</a>, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of <a href="/wiki/Calculus" title="Calculus">calculus</a>,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> the other being <a href="/wiki/Derivative" title="Derivative">differentiation</a>. Integration was initially used to solve problems in mathematics and <a href="/wiki/Physics" title="Physics">physics</a>, such as finding the area under a curve, or determining displacement from velocity. Usage of integration expanded to a wide variety of scientific fields thereafter. </p><p>A <b>definite integral</b> computes the <a href="/wiki/Signed_area" title="Signed area">signed area</a> of the region in the plane that is bounded by the <a href="/wiki/Graph_of_a_function" title="Graph of a function">graph</a> of a given <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> between two points in the <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real line</a>. Conventionally, areas above the horizontal <a href="/wiki/Axis_(mathematics)" class="mw-redirect mw-disambig" title="Axis (mathematics)">axis</a> of the plane are positive while areas below are negative. Integrals also refer to the concept of an <i><a href="/wiki/Antiderivative" title="Antiderivative">antiderivative</a></i>, a function whose <a href="/wiki/Derivative" title="Derivative">derivative</a> is the given function; in this case, they are also called <i>indefinite integrals</i>. The <a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">fundamental theorem of calculus</a> relates definite integration to differentiation and provides a method to compute the definite integral of a function when its antiderivative is known; differentiation and integration are <a href="/wiki/Inverse_function" title="Inverse function">inverse</a> operations. </p><p>Although methods of calculating areas and volumes dated from <a href="/wiki/Ancient_Greek_mathematics" class="mw-redirect" title="Ancient Greek mathematics">ancient Greek mathematics</a>, the principles of integration were formulated independently by <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> and <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a> in the late 17th century, who thought of the area under a curve as an infinite sum of rectangles of <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimal</a> width. <a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a> later gave a rigorous definition of integrals, which is based on a limiting procedure that approximates the area of a <a href="/wiki/Curvilinear_coordinates" title="Curvilinear coordinates">curvilinear</a> region by breaking the region into infinitesimally thin vertical slabs. In the early 20th century, <a href="/wiki/Henri_Lebesgue" title="Henri Lebesgue">Henri Lebesgue</a> generalized Riemann's formulation by introducing what is now referred to as the <a href="/wiki/Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue integral</a>; it is more general than Riemann's in the sense that a wider class of functions are Lebesgue-integrable. </p><p>Integrals may be generalized depending on the type of the function as well as the <a href="/wiki/Domain_(mathematical_analysis)" title="Domain (mathematical analysis)">domain</a> over which the integration is performed. For example, a <a href="/wiki/Line_integral" title="Line integral">line integral</a> is defined for functions of two or more variables, and the <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a> of integration is replaced by a curve connecting two points in space. In a <a href="/wiki/Surface_integral" title="Surface integral">surface integral</a>, the curve is replaced by a piece of a <a href="/wiki/Surface_(mathematics)" title="Surface (mathematics)">surface</a> in <a href="/wiki/Three-dimensional_space" title="Three-dimensional space">three-dimensional space</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/History_of_calculus" title="History of calculus">History of calculus</a></div> <div class="mw-heading mw-heading3"><h3 id="Pre-calculus_integration">Pre-calculus integration</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=2" title="Edit section: Pre-calculus integration"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The first documented systematic technique capable of determining integrals is the <a href="/wiki/Method_of_exhaustion" title="Method of exhaustion">method of exhaustion</a> of the <a href="/wiki/Ancient_Greece" title="Ancient Greece">ancient Greek</a> astronomer <a href="/wiki/Eudoxus_of_Cnidus" title="Eudoxus of Cnidus">Eudoxus</a> and philosopher <a href="/wiki/Democritus" title="Democritus">Democritus</a> (<i>ca.</i> 370 BC), which sought to find areas and volumes by breaking them up into an infinite number of divisions for which the area or volume was known.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> This method was further developed and employed by <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> in the 3rd century BC and used to calculate the <a href="/wiki/Area_of_a_circle" title="Area of a circle">area of a circle</a>, the <a href="/wiki/Surface_area" title="Surface area">surface area</a> and <a href="/wiki/Volume" title="Volume">volume</a> of a <a href="/wiki/Sphere" title="Sphere">sphere</a>, area of an <a href="/wiki/Ellipse" title="Ellipse">ellipse</a>, the area under a <a href="/wiki/Parabola" title="Parabola">parabola</a>, the volume of a segment of a <a href="/wiki/Paraboloid" title="Paraboloid">paraboloid</a> of revolution, the volume of a segment of a <a href="/wiki/Hyperboloid" title="Hyperboloid">hyperboloid</a> of revolution, and the area of a <a href="/wiki/Spiral" title="Spiral">spiral</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>A similar method was independently developed in <a href="/wiki/China" title="China">China</a> around the 3rd century AD by <a href="/wiki/Liu_Hui" title="Liu Hui">Liu Hui</a>, who used it to find the area of the circle. This method was later used in the 5th century by Chinese father-and-son mathematicians <a href="/wiki/Zu_Chongzhi" title="Zu Chongzhi">Zu Chongzhi</a> and <a href="/wiki/Zu_Geng_(mathematician)" class="mw-redirect" title="Zu Geng (mathematician)">Zu Geng</a> to find the volume of a sphere.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>In the Middle East, Hasan Ibn al-Haytham, Latinized as <a href="/wiki/Alhazen" class="mw-redirect" title="Alhazen">Alhazen</a> (<span title="circa">c.</span><span style="white-space:nowrap;"> 965</span> – c.<span style="white-space:nowrap;"> 1040</span> AD) derived a formula for the sum of <a href="/wiki/Fourth_power" title="Fourth power">fourth powers</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Alhazen determined the equations to calculate the area enclosed by the curve represented by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=x^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=x^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7671c58f963c43ee6c236f3e3ed6c54a9c4bfdae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.672ex; height:3.009ex;" alt="{\displaystyle y=x^{k}}"></span> (which translates to the integral <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int x^{k}\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int x^{k}\,dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c636c2d61e234c5306b1682165ab4930b76da88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:7.932ex; height:5.676ex;" alt="{\displaystyle \int x^{k}\,dx}"></span> in contemporary notation), for any given non-negative integer value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> He used the results to carry out what would now be called an integration of this function, where the formulae for the sums of integral squares and fourth powers allowed him to calculate the volume of a <a href="/wiki/Paraboloid" title="Paraboloid">paraboloid</a>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>The next significant advances in integral calculus did not begin to appear until the 17th century. At this time, the work of <a href="/wiki/Bonaventura_Cavalieri" title="Bonaventura Cavalieri">Cavalieri</a> with his <a href="/wiki/Method_of_indivisibles" class="mw-redirect" title="Method of indivisibles">method of indivisibles</a>, and work by <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Fermat</a>, began to lay the foundations of modern calculus,<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> with Cavalieri computing the integrals of <span class="texhtml"><i>x</i><sup><i>n</i></sup></span> up to degree <span class="texhtml"><i>n</i> = 9</span> in <a href="/wiki/Cavalieri%27s_quadrature_formula" title="Cavalieri's quadrature formula">Cavalieri's quadrature formula</a>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> The case <i>n</i> = −1 required the invention of a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a>, the <a href="/wiki/Hyperbolic_logarithm" class="mw-redirect" title="Hyperbolic logarithm">hyperbolic logarithm</a>, achieved by <a href="/wiki/Quadrature_(mathematics)" class="mw-redirect" title="Quadrature (mathematics)">quadrature</a> of the <a href="/wiki/Hyperbola" title="Hyperbola">hyperbola</a> in 1647. </p><p>Further steps were made in the early 17th century by <a href="/wiki/Isaac_Barrow" title="Isaac Barrow">Barrow</a> and <a href="/wiki/Evangelista_Torricelli" title="Evangelista Torricelli">Torricelli</a>, who provided the first hints of a connection between integration and <a href="/wiki/Differential_calculus" title="Differential calculus">differentiation</a>. Barrow provided the first proof of the <a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">fundamental theorem of calculus</a>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> <a href="/wiki/John_Wallis" title="John Wallis">Wallis</a> generalized Cavalieri's method, computing integrals of <span class="texhtml mvar" style="font-style:italic;">x</span> to a general power, including negative powers and fractional powers.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Leibniz_and_Newton">Leibniz and Newton</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=3" title="Edit section: Leibniz and Newton"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The major advance in integration came in the 17th century with the independent discovery of the <a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">fundamental theorem of calculus</a> by <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Leibniz</a> and <a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> The theorem demonstrates a connection between integration and differentiation. This connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the fundamental theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the comprehensive mathematical framework that both Leibniz and Newton developed. Given the name infinitesimal calculus, it allowed for precise analysis of functions with continuous domains. This framework eventually became modern <a href="/wiki/Calculus" title="Calculus">calculus</a>, whose notation for integrals is drawn directly from the work of Leibniz. </p> <div class="mw-heading mw-heading3"><h3 id="Formalization">Formalization</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=4" title="Edit section: Formalization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>While Newton and Leibniz provided a systematic approach to integration, their work lacked a degree of <a href="/wiki/Rigor#Mathematical_rigour" class="mw-redirect" title="Rigor">rigour</a>. <a href="/wiki/George_Berkeley" title="George Berkeley">Bishop Berkeley</a> memorably attacked the vanishing increments used by Newton, calling them "<a href="/wiki/The_Analyst#Content" title="The Analyst">ghosts of departed quantities</a>".<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Calculus acquired a firmer footing with the development of <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">limits</a>. Integration was first rigorously formalized, using limits, by <a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Riemann</a>.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Although all bounded <a href="/wiki/Piecewise" class="mw-redirect" title="Piecewise">piecewise</a> continuous functions are Riemann-integrable on a bounded interval, subsequently more general functions were considered—particularly in the context of <a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a>—to which Riemann's definition does not apply, and <a href="/wiki/Henri_Lebesgue" title="Henri Lebesgue">Lebesgue</a> formulated a <a href="#Lebesgue_integral">different definition of integral</a>, founded in <a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">measure theory</a> (a subfield of <a href="/wiki/Real_analysis" title="Real analysis">real analysis</a>). Other definitions of integral, extending Riemann's and Lebesgue's approaches, were proposed. These approaches based on the real number system are the ones most common today, but alternative approaches exist, such as a definition of integral as the <a href="/wiki/Standard_part" class="mw-redirect" title="Standard part">standard part</a> of an infinite Riemann sum, based on the <a href="/wiki/Hyperreal_number" title="Hyperreal number">hyperreal number</a> system. </p> <div class="mw-heading mw-heading3"><h3 id="Historical_notation">Historical notation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=5" title="Edit section: Historical notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The notation for the indefinite integral was introduced by <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a> in 1675.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> He adapted the <a href="/wiki/Integral_symbol" title="Integral symbol">integral symbol</a>, <b>∫</b>, from the letter <i>ſ</i> (<a href="/wiki/Long_s" title="Long s">long s</a>), standing for <i>summa</i> (written as <i>ſumma</i>; Latin for "sum" or "total"). The modern notation for the definite integral, with limits above and below the integral sign, was first used by <a href="/wiki/Joseph_Fourier" title="Joseph Fourier">Joseph Fourier</a> in <i>Mémoires</i> of the French Academy around 1819–1820, reprinted in his book of 1822.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> used a small vertical bar above a variable to indicate integration, or placed the variable inside a box. The vertical bar was easily confused with <span class="texhtml"><span class="sfrac nowrap;"><span style="display:none; display:inline-block; text-align:center;"><span style="display:block; line-height:0.8em; font-size:70%;"><b>.</b></span><span style="display:block; line-height:1em;"><i>x</i></span></span></span></span> or <span class="texhtml"><i>x</i>′</span>, which are used to indicate differentiation, and the box notation was difficult for printers to reproduce, so these notations were not widely adopted.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="First_use_of_the_term">First use of the term</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=6" title="Edit section: First use of the term"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The term was first printed in Latin by <a href="/wiki/Jacob_Bernoulli" title="Jacob Bernoulli">Jacob Bernoulli</a> in 1690: "Ergo et horum Integralia aequantur".<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Terminology_and_notation">Terminology and notation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=7" title="Edit section: Terminology and notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In general, the integral of a <a href="/wiki/Real-valued_function" title="Real-valued function">real-valued function</a> <span class="texhtml"><i>f</i>(<i>x</i>)</span> with respect to a real variable <span class="texhtml"><i>x</i></span> on an interval <span class="texhtml">[<i>a</i>, <i>b</i>]</span> is written as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}f(x)\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)\,dx.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/249ee3fdbced31dfc328ff357f67bb134ca66b8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.786ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(x)\,dx.}"></span></dd></dl> <p>The integral sign <span class="texhtml">∫</span> represents integration. The symbol <span class="texhtml"><i>dx</i></span>, called the <a href="/wiki/Differential_(infinitesimal)" class="mw-redirect" title="Differential (infinitesimal)">differential</a> of the variable <span class="texhtml"><i>x</i></span>, indicates that the variable of integration is <span class="texhtml"><i>x</i></span>. The function <span class="texhtml"><i>f</i>(<i>x</i>)</span> is called the integrand, the points <span class="texhtml"><i>a</i></span> and <span class="texhtml"><i>b</i></span> are called the limits (or bounds) of integration, and the integral is said to be over the interval <span class="texhtml">[<i>a</i>, <i>b</i>]</span>, called the interval of integration.<sup id="cite_ref-:1_19-0" class="reference"><a href="#cite_note-:1-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> A function is said to be <em>integrable</em><span class="anchor" id="Integrable"></span><span class="anchor" id="Integrable_function"></span> if its integral over its domain is finite. If limits are specified, the integral is called a definite integral. </p><p>When the limits are omitted, as in </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int f(x)\,dx,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int f(x)\,dx,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bbd50daec1c6f3767a42b842a9b203d840e9934" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.578ex; height:5.676ex;" alt="{\displaystyle \int f(x)\,dx,}"></span></dd></dl> <p>the integral is called an indefinite integral, which represents a class of functions (the <a href="/wiki/Antiderivative" title="Antiderivative">antiderivative</a>) whose derivative is the integrand.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">fundamental theorem of calculus</a> relates the evaluation of definite integrals to indefinite integrals. There are several extensions of the notation for integrals to encompass integration on unbounded domains and/or in multiple dimensions (see later sections of this article). </p><p>In advanced settings, it is not uncommon to leave out <span class="texhtml"><i>dx</i></span> when only the simple Riemann integral is being used, or the exact type of integral is immaterial. For instance, one might write <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \int _{a}^{b}(c_{1}f+c_{2}g)=c_{1}\int _{a}^{b}f+c_{2}\int _{a}^{b}g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>f</mi> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \int _{a}^{b}(c_{1}f+c_{2}g)=c_{1}\int _{a}^{b}f+c_{2}\int _{a}^{b}g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73dcac73f0c30b213d7df13d5a0416d6069a75b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.526ex; height:3.676ex;" alt="{\textstyle \int _{a}^{b}(c_{1}f+c_{2}g)=c_{1}\int _{a}^{b}f+c_{2}\int _{a}^{b}g}"></span> to express the linearity of the integral, a property shared by the Riemann integral and all generalizations thereof.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Interpretations">Interpretations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=8" title="Edit section: Interpretations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Integral_approximations_J.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Integral_approximations_J.svg/220px-Integral_approximations_J.svg.png" decoding="async" width="220" height="212" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Integral_approximations_J.svg/330px-Integral_approximations_J.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Integral_approximations_J.svg/440px-Integral_approximations_J.svg.png 2x" data-file-width="219" data-file-height="211" /></a><figcaption>Approximations to integral of <span class="texhtml"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>x</i></span></span></span> from 0 to 1, with 5 yellow right endpoint partitions and 10 green left endpoint partitions</figcaption></figure> <p>Integrals appear in many practical situations. For instance, from the length, width and depth of a swimming pool which is rectangular with a flat bottom, one can determine the volume of water it can contain, the area of its surface, and the length of its edge. But if it is oval with a rounded bottom, integrals are required to find exact and rigorous values for these quantities. In each case, one may divide the sought quantity into infinitely many <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimal</a> pieces, then sum the pieces to achieve an accurate approximation. </p><p>As another example, to find the area of the region bounded by the graph of the function <span class="texhtml"><i>f</i>(<i>x</i>) =</span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\sqrt {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\sqrt {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02f01419d50f8331ed8f948d3b0dce7d5bd75950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.266ex; height:2.843ex;" alt="{\textstyle {\sqrt {x}}}"></span> between <span class="texhtml"><i>x</i> = 0</span> and <span class="texhtml"><i>x</i> = 1</span>, one can divide the interval into five pieces (<span class="texhtml">0, 1/5, 2/5, ..., 1</span>), then construct rectangles using the right end height of each piece (thus <span class="texhtml"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">0</span></span>, <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">1/5</span></span>, <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2/5</span></span>, ..., <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">1</span></span></span>) and sum their areas to get the approximation </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\sqrt {\frac {1}{5}}}\left({\frac {1}{5}}-0\right)+{\sqrt {\frac {2}{5}}}\left({\frac {2}{5}}-{\frac {1}{5}}\right)+\cdots +{\sqrt {\frac {5}{5}}}\left({\frac {5}{5}}-{\frac {4}{5}}\right)\approx 0.7497,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </msqrt> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mo>−</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>2</mn> <mn>5</mn> </mfrac> </msqrt> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>5</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>5</mn> <mn>5</mn> </mfrac> </msqrt> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>5</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>5</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <mn>0.7497</mn> <mo>,</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\sqrt {\frac {1}{5}}}\left({\frac {1}{5}}-0\right)+{\sqrt {\frac {2}{5}}}\left({\frac {2}{5}}-{\frac {1}{5}}\right)+\cdots +{\sqrt {\frac {5}{5}}}\left({\frac {5}{5}}-{\frac {4}{5}}\right)\approx 0.7497,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07cee59815b00288f73cbdd9a9d9b4a3ba331c89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:60.856ex; height:4.843ex;" alt="{\displaystyle \textstyle {\sqrt {\frac {1}{5}}}\left({\frac {1}{5}}-0\right)+{\sqrt {\frac {2}{5}}}\left({\frac {2}{5}}-{\frac {1}{5}}\right)+\cdots +{\sqrt {\frac {5}{5}}}\left({\frac {5}{5}}-{\frac {4}{5}}\right)\approx 0.7497,}"></span></dd></dl> <p>which is larger than the exact value. Alternatively, when replacing these subintervals by ones with the left end height of each piece, the approximation one gets is too low: with twelve such subintervals the approximated area is only 0.6203. However, when the number of pieces increases to infinity, it will reach a limit which is the exact value of the area sought (in this case, <span class="texhtml">2/3</span>). One writes </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{1}{\sqrt {x}}\,dx={\frac {2}{3}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{1}{\sqrt {x}}\,dx={\frac {2}{3}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66f03313f6b5693e7d6f14663fca9efa4472e9ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.848ex; height:6.176ex;" alt="{\displaystyle \int _{0}^{1}{\sqrt {x}}\,dx={\frac {2}{3}},}"></span></dd></dl> <p>which means <span class="texhtml">2/3</span> is the result of a weighted sum of function values, <span class="texhtml"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>x</i></span></span></span>, multiplied by the infinitesimal step widths, denoted by <span class="texhtml"><i>dx</i></span>, on the interval <span class="texhtml">[0, 1]</span>. </p> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tnone center"><div class="thumbinner multiimageinner" style="width:608px;max-width:608px"><div class="trow"><div class="theader" style="text-align:center">Darboux sums</div></div><div class="trow"><div class="tsingle" style="width:302px;max-width:302px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Riemann_Integration_and_Darboux_Upper_Sums.gif" class="mw-file-description"><img alt="Upper Darboux sum example" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Riemann_Integration_and_Darboux_Upper_Sums.gif/300px-Riemann_Integration_and_Darboux_Upper_Sums.gif" decoding="async" width="300" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Riemann_Integration_and_Darboux_Upper_Sums.gif/450px-Riemann_Integration_and_Darboux_Upper_Sums.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Riemann_Integration_and_Darboux_Upper_Sums.gif/600px-Riemann_Integration_and_Darboux_Upper_Sums.gif 2x" data-file-width="800" data-file-height="600" /></a></span></div><div class="thumbcaption text-align-center">Darboux upper sums of the function <span class="texhtml"><i>y</i> = <i>x</i><sup>2</sup></span></div></div><div class="tsingle" style="width:302px;max-width:302px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Riemann_Integration_and_Darboux_Lower_Sums.gif" class="mw-file-description"><img alt="Lower Darboux sum example" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Riemann_Integration_and_Darboux_Lower_Sums.gif/300px-Riemann_Integration_and_Darboux_Lower_Sums.gif" decoding="async" width="300" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Riemann_Integration_and_Darboux_Lower_Sums.gif/450px-Riemann_Integration_and_Darboux_Lower_Sums.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Riemann_Integration_and_Darboux_Lower_Sums.gif/600px-Riemann_Integration_and_Darboux_Lower_Sums.gif 2x" data-file-width="800" data-file-height="600" /></a></span></div><div class="thumbcaption text-align-center">Darboux lower sums of the function <span class="texhtml"><i>y</i> = <i>x</i><sup>2</sup></span></div></div></div></div></div> <div class="mw-heading mw-heading2"><h2 id="Formal_definitions">Formal definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=9" title="Edit section: Formal definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:204px;max-width:204px"><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Riemann_sum_convergence.png" class="mw-file-description"><img alt="Riemann sum convergence" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Riemann_sum_convergence.png/200px-Riemann_sum_convergence.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Riemann_sum_convergence.png/300px-Riemann_sum_convergence.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Riemann_sum_convergence.png/400px-Riemann_sum_convergence.png 2x" data-file-width="1260" data-file-height="1260" /></a></span></div><div class="thumbcaption text-align-center">Riemann sums converging</div></div></div></div></div><p>There are many ways of formally defining an integral, not all of which are equivalent. The differences exist mostly to deal with differing special cases which may not be integrable under other definitions, but are also occasionally for pedagogical reasons. The most commonly used definitions are Riemann integrals and Lebesgue integrals. </p><div class="mw-heading mw-heading3"><h3 id="Riemann_integral">Riemann integral</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=10" title="Edit section: Riemann integral"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Riemann_integral" title="Riemann integral">Riemann integral</a></div> <p>The Riemann integral is defined in terms of <a href="/wiki/Riemann_sum" title="Riemann sum">Riemann sums</a> of functions with respect to <i>tagged partitions</i> of an interval.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> A tagged partition of a <a href="/wiki/Closed_interval" class="mw-redirect" title="Closed interval">closed interval</a> <span class="texhtml">[<i>a</i>, <i>b</i>]</span> on the real line is a finite sequence </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=x_{0}\leq t_{1}\leq x_{1}\leq t_{2}\leq x_{2}\leq \cdots \leq x_{n-1}\leq t_{n}\leq x_{n}=b.\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>≤</mo> <mo>⋯</mo> <mo>≤</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≤</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>b</mi> <mo>.</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=x_{0}\leq t_{1}\leq x_{1}\leq t_{2}\leq x_{2}\leq \cdots \leq x_{n-1}\leq t_{n}\leq x_{n}=b.\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a95dc34e72c3da728c59d5f83e148f3e57f100b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:57.164ex; height:2.509ex;" alt="{\displaystyle a=x_{0}\leq t_{1}\leq x_{1}\leq t_{2}\leq x_{2}\leq \cdots \leq x_{n-1}\leq t_{n}\leq x_{n}=b.\,\!}"></span></dd></dl> <p>This partitions the interval <span class="texhtml">[<i>a</i>, <i>b</i>]</span> into <span class="texhtml mvar" style="font-style:italic;">n</span> sub-intervals <span class="texhtml">[<i>x</i><sub><i>i</i>−1</sub>, <i>x</i><sub><i>i</i></sub>]</span> indexed by <span class="texhtml mvar" style="font-style:italic;">i</span>, each of which is "tagged" with a specific point <span class="texhtml"><i>t</i><sub><i>i</i></sub> ∈ [<i>x</i><sub><i>i</i>−1</sub>, <i>x</i><sub><i>i</i></sub>]</span>. A <i>Riemann sum</i> of a function <span class="texhtml mvar" style="font-style:italic;">f</span> with respect to such a tagged partition is defined as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n}f(t_{i})\,\Delta _{i};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}f(t_{i})\,\Delta _{i};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76e9ff892e308950da2bbe3f2ac7b0a51356c6ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.239ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{n}f(t_{i})\,\Delta _{i};}"></span></dd></dl> <p>thus each term of the sum is the area of a rectangle with height equal to the function value at the chosen point of the given sub-interval, and width the same as the width of sub-interval, <span class="texhtml">Δ<sub><i>i</i></sub> = <i>x</i><sub><i>i</i></sub>−<i>x</i><sub><i>i</i>−1</sub></span>. The <i>mesh</i> of such a tagged partition is the width of the largest sub-interval formed by the partition, <span class="texhtml">max<sub><i>i</i>=1...<i>n</i></sub> Δ<sub><i>i</i></sub></span>. The <i>Riemann integral</i> of a function <span class="texhtml mvar" style="font-style:italic;">f</span> over the interval <span class="texhtml">[<i>a</i>, <i>b</i>]</span> is equal to <span class="texhtml mvar" style="font-style:italic;">S</span> if:<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p> <dl><dd>For all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e04ec3670b50384a3ce48aca42e7cc5131a06b12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}"></span> there exists <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/595d5cea06fdcaf2642caf549eda2cfc537958a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.343ex;" alt="{\displaystyle \delta >0}"></span> such that, for any tagged partition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"></span> with mesh less than <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }"></span>,</dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|S-\sum _{i=1}^{n}f(t_{i})\,\Delta _{i}\right|<\varepsilon .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mi>S</mi> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>|</mo> </mrow> <mo><</mo> <mi>ε</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|S-\sum _{i=1}^{n}f(t_{i})\,\Delta _{i}\right|<\varepsilon .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f42275c8f3dabbf0001fa4fa9d59df0300804af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.054ex; height:7.176ex;" alt="{\displaystyle \left|S-\sum _{i=1}^{n}f(t_{i})\,\Delta _{i}\right|<\varepsilon .}"></span></dd></dl> <p>When the chosen tags are the maximum (respectively, minimum) value of the function in each interval, the Riemann sum becomes an upper (respectively, lower) <a href="/wiki/Darboux_integral" title="Darboux integral">Darboux sum</a>, suggesting the close connection between the Riemann integral and the <a href="/wiki/Darboux_integral" title="Darboux integral">Darboux integral</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Lebesgue_integral">Lebesgue integral</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=11" title="Edit section: Lebesgue integral"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue integration</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Lebesgueintegralsimplefunctions_finer-dotted.svg" class="mw-file-description"><img alt="Comparison of Riemann and Lebesgue integrals" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Lebesgueintegralsimplefunctions_finer-dotted.svg/250px-Lebesgueintegralsimplefunctions_finer-dotted.svg.png" decoding="async" width="250" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Lebesgueintegralsimplefunctions_finer-dotted.svg/375px-Lebesgueintegralsimplefunctions_finer-dotted.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Lebesgueintegralsimplefunctions_finer-dotted.svg/500px-Lebesgueintegralsimplefunctions_finer-dotted.svg.png 2x" data-file-width="366" data-file-height="116" /></a><figcaption>Lebesgue integration</figcaption></figure> <p>It is often of interest, both in theory and applications, to be able to pass to the limit under the integral. For instance, a sequence of functions can frequently be constructed that approximate, in a suitable sense, the solution to a problem. Then the integral of the solution function should be the limit of the integrals of the approximations. However, many functions that can be obtained as limits are not Riemann-integrable, and so such limit theorems do not hold with the Riemann integral. Therefore, it is of great importance to have a definition of the integral that allows a wider class of functions to be integrated.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p>Such an integral is the Lebesgue integral, that exploits the following fact to enlarge the class of integrable functions: if the values of a function are rearranged over the domain, the integral of a function should remain the same. Thus <a href="/wiki/Henri_Lebesgue" title="Henri Lebesgue">Henri Lebesgue</a> introduced the integral bearing his name, explaining this integral thus in a letter to <a href="/wiki/Paul_Montel" title="Paul Montel">Paul Montel</a>:<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>I have to pay a certain sum, which I have collected in my pocket. I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. This is the Riemann integral. But I can proceed differently. After I have taken all the money out of my pocket I order the bills and coins according to identical values and then I pay the several heaps one after the other to the creditor. This is my integral.</p></blockquote> <p>As Folland puts it, "To compute the Riemann integral of <span class="texhtml mvar" style="font-style:italic;">f</span>, one partitions the domain <span class="texhtml">[<i>a</i>, <i>b</i>]</span> into subintervals", while in the Lebesgue integral, "one is in effect partitioning the range of <span class="texhtml mvar" style="font-style:italic;">f</span> ".<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> The definition of the Lebesgue integral thus begins with a <a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">measure</a>, μ. In the simplest case, the <a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a> <span class="texhtml"><i>μ</i>(<i>A</i>)</span> of an interval <span class="texhtml"><i>A</i> = [<i>a</i>, <i>b</i>]</span> is its width, <span class="texhtml"><i>b</i> − <i>a</i></span>, so that the Lebesgue integral agrees with the (proper) Riemann integral when both exist.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> In more complicated cases, the sets being measured can be highly fragmented, with no continuity and no resemblance to intervals. </p><p>Using the "partitioning the range of <span class="texhtml mvar" style="font-style:italic;">f</span> " philosophy, the integral of a non-negative function <span class="texhtml"><i>f</i> : <b>R</b> → <b>R</b></span> should be the sum over <span class="texhtml mvar" style="font-style:italic;">t</span> of the areas between a thin horizontal strip between <span class="texhtml"><i>y</i> = <i>t</i></span> and <span class="texhtml"><i>y</i> = <i>t</i> + <i>dt</i></span>. This area is just <span class="texhtml"><i>μ</i>{ <i>x</i> : <i>f</i>(<i>x</i>) > <i>t</i>} <i>dt</i></span>. Let <span class="texhtml"><i>f</i><sup>∗</sup>(<i>t</i>) = <i>μ</i>{ <i>x</i> : <i>f</i>(<i>x</i>) > <i>t</i> }</span>. The Lebesgue integral of <span class="texhtml mvar" style="font-style:italic;">f</span> is then defined by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int f=\int _{0}^{\infty }f^{*}(t)\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>f</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int f=\int _{0}^{\infty }f^{*}(t)\,dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67ab0b364de19ff1c40853e153ff423fd7c47154" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.151ex; height:5.843ex;" alt="{\displaystyle \int f=\int _{0}^{\infty }f^{*}(t)\,dt}"></span></dd></dl> <p>where the integral on the right is an ordinary improper Riemann integral (<span class="texhtml"><i>f</i><span style="padding-left:0.12em;"><sup>∗</sup></span></span> is a strictly decreasing positive function, and therefore has a <a href="/wiki/Well-defined" class="mw-redirect" title="Well-defined">well-defined</a> improper Riemann integral).<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> For a suitable class of functions (the <a href="/wiki/Measurable_function" title="Measurable function">measurable functions</a>) this defines the Lebesgue integral. </p><p>A general measurable function <span class="texhtml mvar" style="font-style:italic;">f</span> is Lebesgue-integrable if the sum of the absolute values of the areas of the regions between the graph of <span class="texhtml mvar" style="font-style:italic;">f</span> and the <span class="texhtml mvar" style="font-style:italic;">x</span>-axis is finite:<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{E}|f|\,d\mu <+\infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>μ</mi> <mo><</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{E}|f|\,d\mu <+\infty .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f768a56292de5a02009fe6e4060bd6c6f12ed61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.622ex; height:5.676ex;" alt="{\displaystyle \int _{E}|f|\,d\mu <+\infty .}"></span></dd></dl> <p>In that case, the integral is, as in the Riemannian case, the difference between the area above the <span class="texhtml mvar" style="font-style:italic;">x</span>-axis and the area below the <span class="texhtml mvar" style="font-style:italic;">x</span>-axis:<sup id="cite_ref-:3_30-0" class="reference"><a href="#cite_note-:3-30"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{E}f\,d\mu =\int _{E}f^{+}\,d\mu -\int _{E}f^{-}\,d\mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mi>f</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>μ</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>μ</mi> <mo>−</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{E}f\,d\mu =\int _{E}f^{+}\,d\mu -\int _{E}f^{-}\,d\mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1950569fcd75f4d509ab0d3659010b05d63e014" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.396ex; height:5.676ex;" alt="{\displaystyle \int _{E}f\,d\mu =\int _{E}f^{+}\,d\mu -\int _{E}f^{-}\,d\mu }"></span></dd></dl> <p>where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{alignedat}{3}&f^{+}(x)&&{}={}\max\{f(x),0\}&&{}={}{\begin{cases}f(x),&{\text{if }}f(x)>0,\\0,&{\text{otherwise,}}\end{cases}}\\&f^{-}(x)&&{}={}\max\{-f(x),0\}&&{}={}{\begin{cases}-f(x),&{\text{if }}f(x)<0,\\0,&{\text{otherwise.}}\end{cases}}\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo movablelimits="true" form="prefix">max</mo> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mtd> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise,</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo movablelimits="true" form="prefix">max</mo> <mo fence="false" stretchy="false">{</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mtd> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo><</mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise.</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{3}&f^{+}(x)&&{}={}\max\{f(x),0\}&&{}={}{\begin{cases}f(x),&{\text{if }}f(x)>0,\\0,&{\text{otherwise,}}\end{cases}}\\&f^{-}(x)&&{}={}\max\{-f(x),0\}&&{}={}{\begin{cases}-f(x),&{\text{if }}f(x)<0,\\0,&{\text{otherwise.}}\end{cases}}\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02e8d66ef704a22d5abbc090506e442b7587fc23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:51.333ex; height:12.509ex;" alt="{\displaystyle {\begin{alignedat}{3}&f^{+}(x)&&{}={}\max\{f(x),0\}&&{}={}{\begin{cases}f(x),&{\text{if }}f(x)>0,\\0,&{\text{otherwise,}}\end{cases}}\\&f^{-}(x)&&{}={}\max\{-f(x),0\}&&{}={}{\begin{cases}-f(x),&{\text{if }}f(x)<0,\\0,&{\text{otherwise.}}\end{cases}}\end{alignedat}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Other_integrals">Other integrals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=12" title="Edit section: Other integrals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Although the Riemann and Lebesgue integrals are the most widely used definitions of the integral, a number of others exist, including: </p> <ul><li>The <a href="/wiki/Darboux_integral" title="Darboux integral">Darboux integral</a>, which is defined by Darboux sums (restricted Riemann sums) yet is equivalent to the <a href="/wiki/Riemann_integral" title="Riemann integral">Riemann integral</a>. A function is Darboux-integrable if and only if it is Riemann-integrable. Darboux integrals have the advantage of being easier to define than Riemann integrals.</li> <li>The <a href="/wiki/Riemann%E2%80%93Stieltjes_integral" title="Riemann–Stieltjes integral">Riemann–Stieltjes integral</a>, an extension of the Riemann integral which integrates with respect to a function as opposed to a variable.</li> <li>The <a href="/wiki/Lebesgue%E2%80%93Stieltjes_integration" title="Lebesgue–Stieltjes integration">Lebesgue–Stieltjes integral</a>, further developed by <a href="/wiki/Johann_Radon" title="Johann Radon">Johann Radon</a>, which generalizes both the Riemann–Stieltjes and Lebesgue integrals.</li> <li>The <a href="/wiki/Daniell_integral" title="Daniell integral">Daniell integral</a>, which subsumes the Lebesgue integral and <a href="/wiki/Lebesgue%E2%80%93Stieltjes_integration" title="Lebesgue–Stieltjes integration">Lebesgue–Stieltjes integral</a> without depending on <a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">measures</a>.</li> <li>The <a href="/wiki/Haar_integral" class="mw-redirect" title="Haar integral">Haar integral</a>, used for integration on locally compact topological groups, introduced by <a href="/wiki/Alfr%C3%A9d_Haar" title="Alfréd Haar">Alfréd Haar</a> in 1933.</li> <li>The <a href="/wiki/Henstock%E2%80%93Kurzweil_integral" title="Henstock–Kurzweil integral">Henstock–Kurzweil integral</a>, variously defined by <a href="/wiki/Arnaud_Denjoy" title="Arnaud Denjoy">Arnaud Denjoy</a>, <a href="/wiki/Oskar_Perron" title="Oskar Perron">Oskar Perron</a>, and (most elegantly, as the gauge integral) <a href="/wiki/Jaroslav_Kurzweil" title="Jaroslav Kurzweil">Jaroslav Kurzweil</a>, and developed by <a href="/wiki/Ralph_Henstock" title="Ralph Henstock">Ralph Henstock</a>.</li> <li>The <a href="/wiki/It%C3%B4_integral" class="mw-redirect" title="Itô integral">Itô integral</a> and <a href="/wiki/Stratonovich_integral" title="Stratonovich integral">Stratonovich integral</a>, which define integration with respect to <a href="/wiki/Semimartingale" title="Semimartingale">semimartingales</a> such as <a href="/wiki/Wiener_process" title="Wiener process">Brownian motion</a>.</li> <li>The <a href="/wiki/Young_integral" class="mw-redirect" title="Young integral">Young integral</a>, which is a kind of Riemann–Stieltjes integral with respect to certain functions of <a href="/wiki/Bounded_variation" title="Bounded variation">unbounded variation</a>.</li> <li>The <a href="/wiki/Rough_path" title="Rough path">rough path</a> integral, which is defined for functions equipped with some additional "rough path" structure and generalizes stochastic integration against both <a href="/wiki/Semimartingale" title="Semimartingale">semimartingales</a> and processes such as the <a href="/wiki/Fractional_Brownian_motion" title="Fractional Brownian motion">fractional Brownian motion</a>.</li> <li>The <a href="/wiki/Choquet_integral" title="Choquet integral">Choquet integral</a>, a subadditive or superadditive integral created by the French mathematician Gustave Choquet in 1953.</li> <li>The <a href="/wiki/Bochner_integral" title="Bochner integral">Bochner integral</a>, an extension of the Lebesgue integral to a more general class of functions, namely, those with a domain that is a <a href="/wiki/Banach_space" title="Banach space">Banach space</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=13" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Linearity">Linearity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=14" title="Edit section: Linearity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The collection of Riemann-integrable functions on a closed interval <span class="texhtml">[<i>a</i>, <i>b</i>]</span> forms a <a href="/wiki/Vector_space" title="Vector space">vector space</a> under the operations of <a href="/wiki/Pointwise_addition" class="mw-redirect" title="Pointwise addition">pointwise addition</a> and multiplication by a scalar, and the operation of integration </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\mapsto \int _{a}^{b}f(x)\;dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">↦</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\mapsto \int _{a}^{b}f(x)\;dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e75c74f0d1966e467b8dee4b36b9a14d00248bda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.29ex; height:6.343ex;" alt="{\displaystyle f\mapsto \int _{a}^{b}f(x)\;dx}"></span></dd></dl> <p>is a <a href="/wiki/Linear_functional" class="mw-redirect" title="Linear functional">linear functional</a> on this vector space. Thus, the collection of integrable functions is closed under taking <a href="/wiki/Linear_combination" title="Linear combination">linear combinations</a>, and the integral of a linear combination is the linear combination of the integrals:<sup id="cite_ref-:0_31-0" class="reference"><a href="#cite_note-:0-31"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}(\alpha f+\beta g)(x)\,dx=\alpha \int _{a}^{b}f(x)\,dx+\beta \int _{a}^{b}g(x)\,dx.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>α</mi> <mi>f</mi> <mo>+</mo> <mi>β</mi> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>α</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>+</mo> <mi>β</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}(\alpha f+\beta g)(x)\,dx=\alpha \int _{a}^{b}f(x)\,dx+\beta \int _{a}^{b}g(x)\,dx.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26dd4e7559e6708844575862a2770422b4fc0ebb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:52.02ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}(\alpha f+\beta g)(x)\,dx=\alpha \int _{a}^{b}f(x)\,dx+\beta \int _{a}^{b}g(x)\,dx.\,}"></span></dd></dl> <p>Similarly, the set of <a href="/wiki/Real_number" title="Real number">real</a>-valued Lebesgue-integrable functions on a given <a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">measure space</a> <span class="texhtml mvar" style="font-style:italic;">E</span> with measure <span class="texhtml mvar" style="font-style:italic;">μ</span> is closed under taking linear combinations and hence form a vector space, and the Lebesgue integral </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\mapsto \int _{E}f\,d\mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">↦</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mi>f</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\mapsto \int _{E}f\,d\mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9acad9829c6f1b994440d268dcf7fb3da77ee22d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.343ex; height:5.676ex;" alt="{\displaystyle f\mapsto \int _{E}f\,d\mu }"></span></dd></dl> <p>is a linear functional on this vector space, so that:<sup id="cite_ref-:3_30-1" class="reference"><a href="#cite_note-:3-30"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{E}(\alpha f+\beta g)\,d\mu =\alpha \int _{E}f\,d\mu +\beta \int _{E}g\,d\mu .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>α</mi> <mi>f</mi> <mo>+</mo> <mi>β</mi> <mi>g</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>μ</mi> <mo>=</mo> <mi>α</mi> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mi>f</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>μ</mi> <mo>+</mo> <mi>β</mi> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mi>g</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>μ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{E}(\alpha f+\beta g)\,d\mu =\alpha \int _{E}f\,d\mu +\beta \int _{E}g\,d\mu .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/857560e35dc5c14a076966bfda1e5ae972a6c746" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:40.567ex; height:5.676ex;" alt="{\displaystyle \int _{E}(\alpha f+\beta g)\,d\mu =\alpha \int _{E}f\,d\mu +\beta \int _{E}g\,d\mu .}"></span></dd></dl> <p>More generally, consider the vector space of all <a href="/wiki/Measurable_function" title="Measurable function">measurable functions</a> on a measure space <span class="texhtml">(<i>E</i>,<i>μ</i>)</span>, taking values in a <a href="/wiki/Locally_compact_space" title="Locally compact space">locally compact</a> <a href="/wiki/Complete_metric_space" title="Complete metric space">complete</a> <a href="/wiki/Topological_vector_space" title="Topological vector space">topological vector space</a> <span class="texhtml mvar" style="font-style:italic;">V</span> over a locally compact <a href="/wiki/Topological_ring" title="Topological ring">topological field</a> <span class="texhtml"><i>K</i>, <i>f</i> : <i>E</i> → <i>V</i></span>. Then one may define an abstract integration map assigning to each function <span class="texhtml mvar" style="font-style:italic;">f</span> an element of <span class="texhtml mvar" style="font-style:italic;">V</span> or the symbol <span class="texhtml"><i>∞</i></span>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\mapsto \int _{E}f\,d\mu ,\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">↦</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mi>f</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>μ</mi> <mo>,</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\mapsto \int _{E}f\,d\mu ,\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/689c556fb9428a1728bc2c3519c60d2d00671cc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.377ex; height:5.676ex;" alt="{\displaystyle f\mapsto \int _{E}f\,d\mu ,\,}"></span></dd></dl> <p>that is compatible with linear combinations.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> In this situation, the linearity holds for the subspace of functions whose integral is an element of <span class="texhtml mvar" style="font-style:italic;">V</span> (i.e. "finite"). The most important special cases arise when <span class="texhtml mvar" style="font-style:italic;">K</span> is <span class="texhtml"><b>R</b></span>, <span class="texhtml"><b>C</b></span>, or a finite extension of the field <span class="texhtml"><b>Q</b><sub><i>p</i></sub></span> of <a href="/wiki/P-adic_number" title="P-adic number">p-adic numbers</a>, and <span class="texhtml mvar" style="font-style:italic;">V</span> is a finite-dimensional vector space over <span class="texhtml mvar" style="font-style:italic;">K</span>, and when <span class="texhtml"><i>K</i> = <b>C</b></span> and <span class="texhtml mvar" style="font-style:italic;">V</span> is a complex <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a>. </p><p>Linearity, together with some natural continuity properties and normalization for a certain class of "simple" functions, may be used to give an alternative definition of the integral. This is the approach of <a href="/wiki/Daniell_integral" title="Daniell integral">Daniell</a> for the case of real-valued functions on a set <span class="texhtml mvar" style="font-style:italic;">X</span>, generalized by <a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Nicolas Bourbaki</a> to functions with values in a locally compact topological vector space. See <a href="#CITEREFHildebrandt1953">Hildebrandt 1953</a> for an axiomatic characterization of the integral. </p> <div class="mw-heading mw-heading3"><h3 id="Inequalities">Inequalities</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=15" title="Edit section: Inequalities"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A number of general inequalities hold for Riemann-integrable <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a> defined on a <a href="/wiki/Closed_set" title="Closed set">closed</a> and <a href="/wiki/Bounded_set" title="Bounded set">bounded</a> <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a> <span class="texhtml">[<i>a</i>, <i>b</i>]</span> and can be generalized to other notions of integral (Lebesgue and Daniell). </p> <ul><li><i>Upper and lower bounds.</i> An integrable function <span class="texhtml mvar" style="font-style:italic;">f</span> on <span class="texhtml">[<i>a</i>, <i>b</i>]</span>, is necessarily <a href="/wiki/Bounded_function" title="Bounded function">bounded</a> on that interval. Thus there are <a href="/wiki/Real_number" title="Real number">real numbers</a> <span class="texhtml mvar" style="font-style:italic;">m</span> and <span class="texhtml mvar" style="font-style:italic;">M</span> so that <span class="texhtml"><i>m</i> ≤ <i>f</i> (<i>x</i>) ≤ <i>M</i></span> for all <span class="texhtml mvar" style="font-style:italic;">x</span> in <span class="texhtml">[<i>a</i>, <i>b</i>]</span>. Since the lower and upper sums of <span class="texhtml mvar" style="font-style:italic;">f</span> over <span class="texhtml">[<i>a</i>, <i>b</i>]</span> are therefore bounded by, respectively, <span class="texhtml"><i>m</i>(<i>b</i> − <i>a</i>)</span> and <span class="texhtml"><i>M</i>(<i>b</i> − <i>a</i>)</span>, it follows that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m(b-a)\leq \int _{a}^{b}f(x)\,dx\leq M(b-a).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>≤</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m(b-a)\leq \int _{a}^{b}f(x)\,dx\leq M(b-a).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/782f800b43b9db1bdfa991f143322d67185f991e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.22ex; height:6.343ex;" alt="{\displaystyle m(b-a)\leq \int _{a}^{b}f(x)\,dx\leq M(b-a).}"></span></li> <li><i>Inequalities between functions.</i><sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> If <span class="texhtml"><i>f</i>(<i>x</i>) ≤ <i>g</i>(<i>x</i>)</span> for each <span class="texhtml mvar" style="font-style:italic;">x</span> in <span class="texhtml">[<i>a</i>, <i>b</i>]</span> then each of the upper and lower sums of <span class="texhtml mvar" style="font-style:italic;">f</span> is bounded above by the upper and lower sums, respectively, of <span class="texhtml mvar" style="font-style:italic;">g</span>. Thus <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}f(x)\,dx\leq \int _{a}^{b}g(x)\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>≤</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)\,dx\leq \int _{a}^{b}g(x)\,dx.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0a4c6eb7a2f260ba880a015d890ba94e243c164" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.861ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(x)\,dx\leq \int _{a}^{b}g(x)\,dx.}"></span> This is a generalization of the above inequalities, as <span class="texhtml"><i>M</i>(<i>b</i> − <i>a</i>)</span> is the integral of the constant function with value <span class="texhtml mvar" style="font-style:italic;">M</span> over <span class="texhtml">[<i>a</i>, <i>b</i>]</span>. In addition, if the inequality between functions is strict, then the inequality between integrals is also strict. That is, if <span class="texhtml"><i>f</i>(<i>x</i>) < <i>g</i>(<i>x</i>)</span> for each <span class="texhtml mvar" style="font-style:italic;">x</span> in <span class="texhtml">[<i>a</i>, <i>b</i>]</span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}f(x)\,dx<\int _{a}^{b}g(x)\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo><</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)\,dx<\int _{a}^{b}g(x)\,dx.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29b6c34b1e601a3388ccd4988d4e5d3014dc62c0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.861ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(x)\,dx<\int _{a}^{b}g(x)\,dx.}"></span></li> <li><i>Subintervals.</i> If <span class="texhtml">[<i>c</i>, <i>d</i>]</span> is a subinterval of <span class="texhtml">[<i>a</i>, <i>b</i>]</span> and <span class="texhtml"><i>f</i> (<i>x</i>)</span> is non-negative for all <span class="texhtml mvar" style="font-style:italic;">x</span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{c}^{d}f(x)\,dx\leq \int _{a}^{b}f(x)\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>≤</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{c}^{d}f(x)\,dx\leq \int _{a}^{b}f(x)\,dx.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b525b803be2260364819821bc30394eb10795d2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.178ex; height:6.343ex;" alt="{\displaystyle \int _{c}^{d}f(x)\,dx\leq \int _{a}^{b}f(x)\,dx.}"></span></li> <li><i>Products and absolute values of functions.</i> If <span class="texhtml mvar" style="font-style:italic;">f</span> and <span class="texhtml mvar" style="font-style:italic;">g</span> are two functions, then we may consider their <a href="/wiki/Pointwise_product" class="mw-redirect" title="Pointwise product">pointwise products</a> and powers, and <a href="/wiki/Absolute_value" title="Absolute value">absolute values</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (fg)(x)=f(x)g(x),\;f^{2}(x)=(f(x))^{2},\;|f|(x)=|f(x)|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thickmathspace" /> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (fg)(x)=f(x)g(x),\;f^{2}(x)=(f(x))^{2},\;|f|(x)=|f(x)|.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bef28f306593eb8eaaba0e365ff005befc8eb5d8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:53.533ex; height:3.176ex;" alt="{\displaystyle (fg)(x)=f(x)g(x),\;f^{2}(x)=(f(x))^{2},\;|f|(x)=|f(x)|.}"></span> If <span class="texhtml mvar" style="font-style:italic;">f</span> is Riemann-integrable on <span class="texhtml">[<i>a</i>, <i>b</i>]</span> then the same is true for <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>f</i></span>|</span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\int _{a}^{b}f(x)\,dx\right|\leq \int _{a}^{b}|f(x)|\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\int _{a}^{b}f(x)\,dx\right|\leq \int _{a}^{b}|f(x)|\,dx.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d273e267ac96790ce130211f3135e6cad0a3c31" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:28.611ex; height:6.843ex;" alt="{\displaystyle \left|\int _{a}^{b}f(x)\,dx\right|\leq \int _{a}^{b}|f(x)|\,dx.}"></span> Moreover, if <span class="texhtml mvar" style="font-style:italic;">f</span> and <span class="texhtml mvar" style="font-style:italic;">g</span> are both Riemann-integrable then <span class="texhtml"><i>fg</i></span> is also Riemann-integrable, and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\int _{a}^{b}(fg)(x)\,dx\right)^{2}\leq \left(\int _{a}^{b}f(x)^{2}\,dx\right)\left(\int _{a}^{b}g(x)^{2}\,dx\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>f</mi> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≤</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\int _{a}^{b}(fg)(x)\,dx\right)^{2}\leq \left(\int _{a}^{b}f(x)^{2}\,dx\right)\left(\int _{a}^{b}g(x)^{2}\,dx\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc9e4809b17198083e1940bd41feed64957d2147" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:53.739ex; height:6.843ex;" alt="{\displaystyle \left(\int _{a}^{b}(fg)(x)\,dx\right)^{2}\leq \left(\int _{a}^{b}f(x)^{2}\,dx\right)\left(\int _{a}^{b}g(x)^{2}\,dx\right).}"></span> This inequality, known as the <a href="/wiki/Cauchy%E2%80%93Schwarz_inequality" title="Cauchy–Schwarz inequality">Cauchy–Schwarz inequality</a>, plays a prominent role in <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> theory, where the left hand side is interpreted as the <a href="/wiki/Inner_product_space" title="Inner product space">inner product</a> of two <a href="/wiki/Square-integrable_function" title="Square-integrable function">square-integrable</a> functions <span class="texhtml mvar" style="font-style:italic;">f</span> and <span class="texhtml mvar" style="font-style:italic;">g</span> on the interval <span class="texhtml">[<i>a</i>, <i>b</i>]</span>.</li> <li><i>Hölder's inequality</i>.<sup id="cite_ref-:4_34-0" class="reference"><a href="#cite_note-:4-34"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> Suppose that <span class="texhtml mvar" style="font-style:italic;">p</span> and <span class="texhtml mvar" style="font-style:italic;">q</span> are two real numbers, <span class="texhtml">1 ≤ <i>p</i>, <i>q</i> ≤ ∞</span> with <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"><i>p</i></span></span>⁠</span> + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"><i>q</i></span></span>⁠</span> = 1</span>, and <span class="texhtml mvar" style="font-style:italic;">f</span> and <span class="texhtml mvar" style="font-style:italic;">g</span> are two Riemann-integrable functions. Then the functions <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>f</i></span>|<sup><i>p</i></sup></span> and <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>g</i></span>|<sup><i>q</i></sup></span> are also integrable and the following <a href="/wiki/H%C3%B6lder%27s_inequality" title="Hölder's inequality">Hölder's inequality</a> holds: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\int f(x)g(x)\,dx\right|\leq \left(\int \left|f(x)\right|^{p}\,dx\right)^{1/p}\left(\int \left|g(x)\right|^{q}\,dx\right)^{1/q}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mo>∫</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <msup> <mrow> <mo>(</mo> <mrow> <mo>∫</mo> <msup> <mrow> <mo>|</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mo>∫</mo> <msup> <mrow> <mo>|</mo> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>q</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\int f(x)g(x)\,dx\right|\leq \left(\int \left|f(x)\right|^{p}\,dx\right)^{1/p}\left(\int \left|g(x)\right|^{q}\,dx\right)^{1/q}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd7fff3624f6c19b3d82b11455ee2f22be0e1f7c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:55.737ex; height:6.676ex;" alt="{\displaystyle \left|\int f(x)g(x)\,dx\right|\leq \left(\int \left|f(x)\right|^{p}\,dx\right)^{1/p}\left(\int \left|g(x)\right|^{q}\,dx\right)^{1/q}.}"></span> For <span class="texhtml"><i>p</i> = <i>q</i> = 2</span>, Hölder's inequality becomes the Cauchy–Schwarz inequality.</li> <li><i>Minkowski inequality</i>.<sup id="cite_ref-:4_34-1" class="reference"><a href="#cite_note-:4-34"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> Suppose that <span class="texhtml"><i>p</i> ≥ 1</span> is a real number and <span class="texhtml mvar" style="font-style:italic;">f</span> and <span class="texhtml mvar" style="font-style:italic;">g</span> are Riemann-integrable functions. Then <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"> <i>f</i> </span>|<sup><i>p</i></sup>, |<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"> <i>g</i> </span>|<sup><i>p</i></sup></span> and <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"> <i>f</i> + <i>g</i> </span>|<sup><i>p</i></sup></span> are also Riemann-integrable and the following <a href="/wiki/Minkowski_inequality" title="Minkowski inequality">Minkowski inequality</a> holds: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\int \left|f(x)+g(x)\right|^{p}\,dx\right)^{1/p}\leq \left(\int \left|f(x)\right|^{p}\,dx\right)^{1/p}+\left(\int \left|g(x)\right|^{p}\,dx\right)^{1/p}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mo>∫</mo> <msup> <mrow> <mo>|</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> </mrow> </msup> <mo>≤</mo> <msup> <mrow> <mo>(</mo> <mrow> <mo>∫</mo> <msup> <mrow> <mo>|</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <mo>∫</mo> <msup> <mrow> <mo>|</mo> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\int \left|f(x)+g(x)\right|^{p}\,dx\right)^{1/p}\leq \left(\int \left|f(x)\right|^{p}\,dx\right)^{1/p}+\left(\int \left|g(x)\right|^{p}\,dx\right)^{1/p}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/912a0477da36636adc618fecc73946dcf0ee6c8e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:68.743ex; height:6.676ex;" alt="{\displaystyle \left(\int \left|f(x)+g(x)\right|^{p}\,dx\right)^{1/p}\leq \left(\int \left|f(x)\right|^{p}\,dx\right)^{1/p}+\left(\int \left|g(x)\right|^{p}\,dx\right)^{1/p}.}"></span> An analogue of this inequality for Lebesgue integral is used in construction of <a href="/wiki/Lp_space" title="Lp space">L<sup>p</sup> spaces</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Conventions">Conventions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=16" title="Edit section: Conventions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In this section, <span class="texhtml mvar" style="font-style:italic;">f</span> is a <a href="/wiki/Real_number" title="Real number">real-valued</a> Riemann-integrable <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a>. The integral </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}f(x)\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)\,dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac02adeed584466d53dee65f3228ad66939eb58b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.139ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(x)\,dx}"></span></dd></dl> <p>over an interval <span class="texhtml">[<i>a</i>, <i>b</i>]</span> is defined if <span class="texhtml"><i>a</i> < <i>b</i></span>. This means that the upper and lower sums of the function <span class="texhtml mvar" style="font-style:italic;">f</span> are evaluated on a partition <span class="texhtml"><i>a</i> = <i>x</i><sub>0</sub> ≤ <i>x</i><sub>1</sub> ≤ . . . ≤ <i>x</i><sub><i>n</i></sub> = <i>b</i></span> whose values <span class="texhtml"><i>x</i><sub><i>i</i></sub></span> are increasing. Geometrically, this signifies that integration takes place "left to right", evaluating <span class="texhtml mvar" style="font-style:italic;">f</span> within intervals <span class="texhtml">[<i>x</i><sub> <i>i</i></sub> , <i>x</i><sub> <i>i</i> +1</sub>]</span> where an interval with a higher index lies to the right of one with a lower index. The values <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span>, the end-points of the <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a>, are called the <a href="/wiki/Limits_of_integration" title="Limits of integration">limits of integration</a> of <span class="texhtml mvar" style="font-style:italic;">f</span>. Integrals can also be defined if <span class="texhtml"><i>a</i> > <i>b</i></span>:<i><sup id="cite_ref-:1_19-1" class="reference"><a href="#cite_note-:1-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup></i> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}f(x)\,dx=-\int _{b}^{a}f(x)\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)\,dx=-\int _{b}^{a}f(x)\,dx.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1f0e0394818f47c41957ff9281eaf3c305495a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.383ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(x)\,dx=-\int _{b}^{a}f(x)\,dx.}"></span></dd></dl> <p>With <span class="texhtml"><i>a</i> = <i>b</i></span>, this implies: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{a}f(x)\,dx=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{a}f(x)\,dx=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f606503473aef1bb7d3ef5ca4e702a7484c5638c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.211ex; height:5.843ex;" alt="{\displaystyle \int _{a}^{a}f(x)\,dx=0.}"></span></dd></dl> <p>The first convention is necessary in consideration of taking integrals over subintervals of <span class="texhtml">[<i>a</i>, <i>b</i>]</span>; the second says that an integral taken over a degenerate interval, or a <a href="/wiki/Point_(geometry)" title="Point (geometry)">point</a>, should be <a href="/wiki/0_(number)" class="mw-redirect" title="0 (number)">zero</a>. One reason for the first convention is that the integrability of <span class="texhtml mvar" style="font-style:italic;">f</span> on an interval <span class="texhtml">[<i>a</i>, <i>b</i>]</span> implies that <span class="texhtml mvar" style="font-style:italic;">f</span> is integrable on any subinterval <span class="texhtml">[<i>c</i>, <i>d</i>]</span>, but in particular integrals have the property that if <span class="texhtml mvar" style="font-style:italic;">c</span> is any <a href="/wiki/Element_(mathematics)" title="Element (mathematics)">element</a> of <span class="texhtml">[<i>a</i>, <i>b</i>]</span>, then:<i><sup id="cite_ref-:0_31-1" class="reference"><a href="#cite_note-:0-31"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup></i> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}f(x)\,dx=\int _{a}^{c}f(x)\,dx+\int _{c}^{b}f(x)\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>+</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)\,dx=\int _{a}^{c}f(x)\,dx+\int _{c}^{b}f(x)\,dx.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55d91ecf0b6d156886defe70f12327d351e385cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:40.01ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(x)\,dx=\int _{a}^{c}f(x)\,dx+\int _{c}^{b}f(x)\,dx.}"></span></dd></dl> <p>With the first convention, the resulting relation </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\int _{a}^{c}f(x)\,dx&{}=\int _{a}^{b}f(x)\,dx-\int _{c}^{b}f(x)\,dx\\&{}=\int _{a}^{b}f(x)\,dx+\int _{b}^{c}f(x)\,dx\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>+</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int _{a}^{c}f(x)\,dx&{}=\int _{a}^{b}f(x)\,dx-\int _{c}^{b}f(x)\,dx\\&{}=\int _{a}^{b}f(x)\,dx+\int _{b}^{c}f(x)\,dx\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d655ef7683fe3dceda35d9566e748c395fc6571d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:40.121ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}\int _{a}^{c}f(x)\,dx&{}=\int _{a}^{b}f(x)\,dx-\int _{c}^{b}f(x)\,dx\\&{}=\int _{a}^{b}f(x)\,dx+\int _{b}^{c}f(x)\,dx\end{aligned}}}"></span></dd></dl> <p>is then well-defined for any cyclic permutation of <span class="texhtml mvar" style="font-style:italic;">a</span>, <span class="texhtml mvar" style="font-style:italic;">b</span>, and <span class="texhtml mvar" style="font-style:italic;">c</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Fundamental_theorem_of_calculus">Fundamental theorem of calculus</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=17" title="Edit section: Fundamental theorem of calculus"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">Fundamental theorem of calculus</a></div> <p>The <i>fundamental theorem of calculus</i> is the statement that <a href="/wiki/Derivative" title="Derivative">differentiation</a> and integration are inverse operations: if a <a href="/wiki/Continuous_function" title="Continuous function">continuous function</a> is first integrated and then differentiated, the original function is retrieved.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> An important consequence, sometimes called the <i>second fundamental theorem of calculus</i>, allows one to compute integrals by using an antiderivative of the function to be integrated.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="First_theorem">First theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=18" title="Edit section: First theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="texhtml mvar" style="font-style:italic;">f</span> be a continuous real-valued function defined on a <a href="/wiki/Interval_(mathematics)#Definitions" title="Interval (mathematics)">closed interval</a> <span class="texhtml">[<i>a</i>, <i>b</i>]</span>. Let <span class="texhtml mvar" style="font-style:italic;">F</span> be the function defined, for all <span class="texhtml mvar" style="font-style:italic;">x</span> in <span class="texhtml">[<i>a</i>, <i>b</i>]</span>, by<sup id="cite_ref-FOOTNOTEMontesinosZizlerZizler2015355_37-0" class="reference"><a href="#cite_note-FOOTNOTEMontesinosZizlerZizler2015355-37"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)=\int _{a}^{x}f(t)\,dt.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)=\int _{a}^{x}f(t)\,dt.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/563036276f93ee00c4afaf3ee6cf668792d07589" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.019ex; height:5.843ex;" alt="{\displaystyle F(x)=\int _{a}^{x}f(t)\,dt.}"></span></dd></dl> <p>Then, <span class="texhtml mvar" style="font-style:italic;">F</span> is continuous on <span class="texhtml">[<i>a</i>, <i>b</i>]</span>, differentiable on the open interval <span class="texhtml">(<i>a</i>, <i>b</i>)</span>, and </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F'(x)=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F'(x)=f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5457591f5410f4bfe3b9c9fa2e50ae665fa2822c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.154ex; height:3.009ex;" alt="{\displaystyle F'(x)=f(x)}"></span></dd></dl> <p>for all <span class="texhtml mvar" style="font-style:italic;">x</span> in <span class="texhtml">(<i>a</i>, <i>b</i>)</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Second_theorem">Second theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=19" title="Edit section: Second theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="texhtml mvar" style="font-style:italic;">f</span> be a real-valued function defined on a <a href="/wiki/Closed_interval" class="mw-redirect" title="Closed interval">closed interval</a> [<span class="texhtml"><i>a</i>, <i>b</i></span>] that admits an <a href="/wiki/Antiderivative" title="Antiderivative">antiderivative</a> <span class="texhtml mvar" style="font-style:italic;">F</span> on <span class="texhtml">[<i>a</i>, <i>b</i>]</span>. That is, <span class="texhtml mvar" style="font-style:italic;">f</span> and <span class="texhtml mvar" style="font-style:italic;">F</span> are functions such that for all <span class="texhtml mvar" style="font-style:italic;">x</span> in <span class="texhtml">[<i>a</i>, <i>b</i>]</span>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=F'(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=F'(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90e48248b0c2565a6f9ba6b20cb4cde93598d5bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.801ex; height:3.009ex;" alt="{\displaystyle f(x)=F'(x).}"></span></dd></dl> <p>If <span class="texhtml mvar" style="font-style:italic;">f</span> is integrable on <span class="texhtml">[<i>a</i>, <i>b</i>]</span> then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}f(x)\,dx=F(b)-F(a).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)\,dx=F(b)-F(a).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f470e7743fda04c3d353a4dee2f441ae454f528" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.052ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(x)\,dx=F(b)-F(a).}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Extensions">Extensions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=20" title="Edit section: Extensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Improper_integrals">Improper integrals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=21" title="Edit section: Improper integrals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Improper_integral" title="Improper integral">Improper integral</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Improper_integral.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Improper_integral.svg/220px-Improper_integral.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Improper_integral.svg/330px-Improper_integral.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/94/Improper_integral.svg/440px-Improper_integral.svg.png 2x" data-file-width="420" data-file-height="420" /></a><figcaption>The <a href="/wiki/Improper_integral" title="Improper integral">improper integral</a><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{\infty }{\frac {dx}{(x+1){\sqrt {x}}}}=\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\infty }{\frac {dx}{(x+1){\sqrt {x}}}}=\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b714edc05d14092854ef3e449cf315ba6e989f3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:20.401ex; height:6.343ex;" alt="{\displaystyle \int _{0}^{\infty }{\frac {dx}{(x+1){\sqrt {x}}}}=\pi }"></span> has unbounded intervals for both domain and range.</figcaption></figure> <p>A "proper" Riemann integral assumes the integrand is defined and finite on a closed and bounded interval, bracketed by the limits of integration. An improper integral occurs when one or more of these conditions is not satisfied. In some cases such integrals may be defined by considering the <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">limit</a> of a <a href="/wiki/Sequence" title="Sequence">sequence</a> of proper <a href="/wiki/Riemann_integral" title="Riemann integral">Riemann integrals</a> on progressively larger intervals. </p><p>If the interval is unbounded, for instance at its upper end, then the improper integral is the limit as that endpoint goes to infinity:<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{\infty }f(x)\,dx=\lim _{b\to \infty }\int _{a}^{b}f(x)\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{\infty }f(x)\,dx=\lim _{b\to \infty }\int _{a}^{b}f(x)\,dx.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9089a514f9e1c1d1dfb26951f80935a63ccce9dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.34ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{\infty }f(x)\,dx=\lim _{b\to \infty }\int _{a}^{b}f(x)\,dx.}"></span></dd></dl> <p>If the integrand is only defined or finite on a half-open interval, for instance <span class="texhtml">(<i>a</i>, <i>b</i>]</span>, then again a limit may provide a finite result:<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}f(x)\,dx=\lim _{\varepsilon \to 0}\int _{a+\epsilon }^{b}f(x)\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>+</mo> <mi>ϵ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)\,dx=\lim _{\varepsilon \to 0}\int _{a+\epsilon }^{b}f(x)\,dx.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82a30070a59eaeb9d9464aa4e2d085202f4b889" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.581ex; height:6.509ex;" alt="{\displaystyle \int _{a}^{b}f(x)\,dx=\lim _{\varepsilon \to 0}\int _{a+\epsilon }^{b}f(x)\,dx.}"></span></dd></dl> <p>That is, the improper integral is the <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">limit</a> of proper integrals as one endpoint of the interval of integration approaches either a specified <a href="/wiki/Real_number" title="Real number">real number</a>, or <span class="texhtml">∞</span>, or <span class="texhtml">−∞</span>. In more complicated cases, limits are required at both endpoints, or at interior points. </p> <div class="mw-heading mw-heading3"><h3 id="Multiple_integration">Multiple integration</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=22" title="Edit section: Multiple integration"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Multiple_integral" title="Multiple integral">Multiple integral</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Volume_under_surface.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Volume_under_surface.png/220px-Volume_under_surface.png" decoding="async" width="220" height="264" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Volume_under_surface.png/330px-Volume_under_surface.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Volume_under_surface.png/440px-Volume_under_surface.png 2x" data-file-width="636" data-file-height="762" /></a><figcaption>Double integral computes volume under a surface <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=f(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=f(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eefb2840000f404c8c0f3f5d6d72f2624854591" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.794ex; height:2.843ex;" alt="{\displaystyle z=f(x,y)}"></span></figcaption></figure> <p>Just as the definite integral of a positive function of one variable represents the <a href="/wiki/Area" title="Area">area</a> of the region between the graph of the function and the <i>x</i>-axis, the <i>double integral</i> of a positive function of two variables represents the <a href="/wiki/Volume" title="Volume">volume</a> of the region between the surface defined by the function and the plane that contains its domain.<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> For example, a function in two dimensions depends on two real variables, <i>x</i> and <i>y</i>, and the integral of a function <i>f</i> over the rectangle <i>R</i> given as the <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a> of two intervals <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=[a,b]\times [c,d]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo>×</mo> <mo stretchy="false">[</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=[a,b]\times [c,d]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3493a3bdcd7bd76960bb3d7b766bd6c6b1c4b3ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.808ex; height:2.843ex;" alt="{\displaystyle R=[a,b]\times [c,d]}"></span> can be written </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{R}f(x,y)\,dA}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{R}f(x,y)\,dA}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a522eea362f7f00960feb1b7ae608fd6182eb1ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.112ex; height:5.676ex;" alt="{\displaystyle \int _{R}f(x,y)\,dA}"></span></dd></dl> <p>where the differential <span class="texhtml"><i>dA</i></span> indicates that integration is taken with respect to area. This <a href="/wiki/Double_integral" class="mw-redirect" title="Double integral">double integral</a> can be defined using <a href="/wiki/Riemann_sum" title="Riemann sum">Riemann sums</a>, and represents the (signed) volume under the graph of <span class="texhtml"><i>z</i> = <i>f</i>(<i>x</i>,<i>y</i>)</span> over the domain <i>R</i>.<sup id="cite_ref-:2_41-0" class="reference"><a href="#cite_note-:2-41"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> Under suitable conditions (e.g., if <i>f</i> is continuous), <a href="/wiki/Fubini%27s_theorem" title="Fubini's theorem">Fubini's theorem</a> states that this integral can be expressed as an equivalent iterated integral<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}\left[\int _{c}^{d}f(x,y)\,dy\right]\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow> <mo>[</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> </mrow> <mo>]</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}\left[\int _{c}^{d}f(x,y)\,dy\right]\,dx.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/256270a484958e5779a7620acb794e26ee9fde16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.519ex; height:6.509ex;" alt="{\displaystyle \int _{a}^{b}\left[\int _{c}^{d}f(x,y)\,dy\right]\,dx.}"></span></dd></dl> <p>This reduces the problem of computing a double integral to computing one-dimensional integrals. Because of this, another notation for the integral over <i>R</i> uses a double integral sign:<sup id="cite_ref-:2_41-1" class="reference"><a href="#cite_note-:2-41"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iint _{R}f(x,y)\,dA.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>A</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iint _{R}f(x,y)\,dA.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/190c3e6f8610e86b6217149de3692b44c1d52253" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.985ex; height:5.676ex;" alt="{\displaystyle \iint _{R}f(x,y)\,dA.}"></span></dd></dl> <p>Integration over more general domains is possible. The integral of a function <i>f</i>, with respect to volume, over an <i>n-</i>dimensional region <i>D</i> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> is denoted by symbols such as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{D}f(\mathbf {x} )d^{n}\mathbf {x} \ =\int _{D}f\,dV.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mtext> </mtext> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mi>f</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{D}f(\mathbf {x} )d^{n}\mathbf {x} \ =\int _{D}f\,dV.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cd16c435cd5581025dde98cc8625d1bb63dbb38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.886ex; height:5.676ex;" alt="{\displaystyle \int _{D}f(\mathbf {x} )d^{n}\mathbf {x} \ =\int _{D}f\,dV.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Line_integrals_and_surface_integrals">Line integrals and surface integrals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=23" title="Edit section: Line integrals and surface integrals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Line_integral" title="Line integral">Line integral</a> and <a href="/wiki/Surface_integral" title="Surface integral">Surface integral</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Line-Integral.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Line-Integral.gif/220px-Line-Integral.gif" decoding="async" width="220" height="202" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/d/d8/Line-Integral.gif 1.5x" data-file-width="300" data-file-height="275" /></a><figcaption>A line integral sums together elements along a curve.</figcaption></figure> <p>The concept of an integral can be extended to more general domains of integration, such as curved lines and surfaces inside higher-dimensional spaces. Such integrals are known as line integrals and surface integrals respectively. These have important applications in physics, as when dealing with <a href="/wiki/Vector_field" title="Vector field">vector fields</a>. </p><p>A <i>line integral</i> (sometimes called a <i>path integral</i>) is an integral where the <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> to be integrated is evaluated along a <a href="/wiki/Curve" title="Curve">curve</a>.<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> Various different line integrals are in use. In the case of a closed curve it is also called a <i>contour integral</i>. </p><p>The function to be integrated may be a <a href="/wiki/Scalar_field" title="Scalar field">scalar field</a> or a <a href="/wiki/Vector_field" title="Vector field">vector field</a>. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly <a href="/wiki/Arc_length" title="Arc length">arc length</a> or, for a vector field, the <a href="/wiki/Inner_product_space" title="Inner product space">scalar product</a> of the vector field with a <a href="/wiki/Differential_(infinitesimal)" class="mw-redirect" title="Differential (infinitesimal)">differential</a> vector in the curve).<sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> This weighting distinguishes the line integral from simpler integrals defined on <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">intervals</a>. Many simple formulas in physics have natural continuous analogs in terms of line integrals; for example, the fact that <a href="/wiki/Mechanical_work" class="mw-redirect" title="Mechanical work">work</a> is equal to <a href="/wiki/Force" title="Force">force</a>, <span class="texhtml"><b>F</b></span>, multiplied by displacement, <span class="texhtml"><b>s</b></span>, may be expressed (in terms of vector quantities) as:<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\mathbf {F} \cdot \mathbf {s} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\mathbf {F} \cdot \mathbf {s} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2102143a354b5c4cfe19a50340a1b568219a49ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.598ex; height:2.176ex;" alt="{\displaystyle W=\mathbf {F} \cdot \mathbf {s} .}"></span></dd></dl> <p>For an object moving along a path <span class="texhtml mvar" style="font-style:italic;"><i>C</i></span> in a <a href="/wiki/Vector_field" title="Vector field">vector field</a> <span class="texhtml"><b>F</b></span> such as an <a href="/wiki/Electric_field" title="Electric field">electric field</a> or <a href="/wiki/Gravitational_field" title="Gravitational field">gravitational field</a>, the total work done by the field on the object is obtained by summing up the differential work done in moving from <span class="texhtml"><b>s</b></span> to <span class="texhtml"><b>s</b> + <i>d</i><b>s</b></span>. This gives the line integral<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\int _{C}\mathbf {F} \cdot d\mathbf {s} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{C}\mathbf {F} \cdot d\mathbf {s} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d831488a8fe259c861dc13cb61643918fd70f56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.975ex; height:5.676ex;" alt="{\displaystyle W=\int _{C}\mathbf {F} \cdot d\mathbf {s} .}"></span></dd></dl> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Surface_integral_illustration.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Surface_integral_illustration.svg/220px-Surface_integral_illustration.svg.png" decoding="async" width="220" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Surface_integral_illustration.svg/330px-Surface_integral_illustration.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/72/Surface_integral_illustration.svg/440px-Surface_integral_illustration.svg.png 2x" data-file-width="512" data-file-height="348" /></a><figcaption>The definition of surface integral relies on splitting the surface into small surface elements.</figcaption></figure> <p>A <i>surface integral</i> generalizes double integrals to integration over a <a href="/wiki/Surface_(mathematics)" title="Surface (mathematics)">surface</a> (which may be a curved set in <a href="/wiki/Space" title="Space">space</a>); it can be thought of as the <a href="/wiki/Multiple_integral" title="Multiple integral">double integral</a> analog of the <a href="/wiki/Line_integral" title="Line integral">line integral</a>. The function to be integrated may be a <a href="/wiki/Scalar_field" title="Scalar field">scalar field</a> or a <a href="/wiki/Vector_field" title="Vector field">vector field</a>. The value of the surface integral is the sum of the field at all points on the surface. This can be achieved by splitting the surface into surface elements, which provide the partitioning for Riemann sums.<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> </p><p>For an example of applications of surface integrals, consider a vector field <span class="texhtml"><b>v</b></span> on a surface <span class="texhtml"><i>S</i></span>; that is, for each point <span class="texhtml"><i>x</i></span> in <span class="texhtml"><i>S</i></span>, <span class="texhtml"><b>v</b>(<i>x</i>)</span> is a vector. Imagine that a fluid flows through <span class="texhtml"><i>S</i></span>, such that <span class="texhtml"><b>v</b>(<i>x</i>)</span> determines the velocity of the fluid at <span class="texhtml mvar" style="font-style:italic;">x</span>. The <a href="/wiki/Flux" title="Flux">flux</a> is defined as the quantity of fluid flowing through <span class="texhtml"><i>S</i></span> in unit amount of time. To find the flux, one need to take the <a href="/wiki/Dot_product" title="Dot product">dot product</a> of <span class="texhtml"><b>v</b></span> with the unit <a href="/wiki/Normal_(geometry)" title="Normal (geometry)">surface normal</a> to <span class="texhtml"><i>S</i></span> at each point, which will give a scalar field, which is integrated over the surface:<sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{S}{\mathbf {v} }\cdot \,d{\mathbf {S} }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mo>⋅</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{S}{\mathbf {v} }\cdot \,d{\mathbf {S} }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27a969275a87437b804ded78fb9b02d57ce1dbbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.797ex; height:5.676ex;" alt="{\displaystyle \int _{S}{\mathbf {v} }\cdot \,d{\mathbf {S} }.}"></span></dd></dl> <p>The fluid flux in this example may be from a physical fluid such as water or air, or from electrical or magnetic flux. Thus surface integrals have applications in physics, particularly with the <a href="/wiki/Classical_theory" class="mw-redirect" title="Classical theory">classical theory</a> of <a href="/wiki/Electromagnetism" title="Electromagnetism">electromagnetism</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Contour_integrals">Contour integrals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=24" title="Edit section: Contour integrals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Contour_integration" title="Contour integration">Contour integration</a></div> <p>In <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a>, the integrand is a <a href="/wiki/Complex-valued_function" class="mw-redirect" title="Complex-valued function">complex-valued function</a> of a complex variable <span class="texhtml mvar" style="font-style:italic;">z</span> instead of a real function of a real variable <span class="texhtml mvar" style="font-style:italic;">x</span>. When a complex function is integrated along a curve <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> in the complex plane, the integral is denoted as follows </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\gamma }f(z)\,dz.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\gamma }f(z)\,dz.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adf4ad465dcdd2fddac974923eaa3ed1913186e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:10.318ex; height:6.009ex;" alt="{\displaystyle \int _{\gamma }f(z)\,dz.}"></span></dd></dl> <p>This is known as a <a href="/wiki/Contour_integral" class="mw-redirect" title="Contour integral">contour integral</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Integrals_of_differential_forms">Integrals of differential forms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=25" title="Edit section: Integrals of differential forms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Differential_form" title="Differential form">Differential form</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Volume_form" title="Volume form">Volume form</a> and <a href="/wiki/Density_on_a_manifold" title="Density on a manifold">Density on a manifold</a></div> <p>A <a href="/wiki/Differential_form" title="Differential form">differential form</a> is a mathematical concept in the fields of <a href="/wiki/Multivariable_calculus" title="Multivariable calculus">multivariable calculus</a>, <a href="/wiki/Differential_topology" title="Differential topology">differential topology</a>, and <a href="/wiki/Tensor" title="Tensor">tensors</a>. Differential forms are organized by degree. For example, a one-form is a weighted sum of the differentials of the coordinates, such as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(x,y,z)\,dx+F(x,y,z)\,dy+G(x,y,z)\,dz}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>+</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> <mo>+</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(x,y,z)\,dx+F(x,y,z)\,dy+G(x,y,z)\,dz}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40dad737a1822d5c7e986c7c240c9d1027a863ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.758ex; height:2.843ex;" alt="{\displaystyle E(x,y,z)\,dx+F(x,y,z)\,dy+G(x,y,z)\,dz}"></span></dd></dl> <p>where <i>E</i>, <i>F</i>, <i>G</i> are functions in three dimensions. A differential one-form can be integrated over an oriented path, and the resulting integral is just another way of writing a line integral. Here the basic differentials <i>dx</i>, <i>dy</i>, <i>dz</i> measure infinitesimal oriented lengths parallel to the three coordinate axes. </p><p>A differential two-form is a sum of the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(x,y,z)\,dx\wedge dy+E(x,y,z)\,dy\wedge dz+F(x,y,z)\,dz\wedge dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>∧</mo> <mi>d</mi> <mi>y</mi> <mo>+</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> <mo>∧</mo> <mi>d</mi> <mi>z</mi> <mo>+</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> <mo>∧</mo> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(x,y,z)\,dx\wedge dy+E(x,y,z)\,dy\wedge dz+F(x,y,z)\,dz\wedge dx.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/190aa2cc6a89d697faf1e75090a09c632a166607" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.373ex; height:2.843ex;" alt="{\displaystyle G(x,y,z)\,dx\wedge dy+E(x,y,z)\,dy\wedge dz+F(x,y,z)\,dz\wedge dx.}"></span></dd></dl> <p>Here the basic two-forms <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dx\wedge dy,dz\wedge dx,dy\wedge dz}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>x</mi> <mo>∧</mo> <mi>d</mi> <mi>y</mi> <mo>,</mo> <mi>d</mi> <mi>z</mi> <mo>∧</mo> <mi>d</mi> <mi>x</mi> <mo>,</mo> <mi>d</mi> <mi>y</mi> <mo>∧</mo> <mi>d</mi> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dx\wedge dy,dz\wedge dx,dy\wedge dz}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c285fb7d96a1408455e9e5510b394a79d50e0cd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.258ex; height:2.509ex;" alt="{\displaystyle dx\wedge dy,dz\wedge dx,dy\wedge dz}"></span> measure oriented areas parallel to the coordinate two-planes. The symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }"></span> denotes the <a href="/wiki/Wedge_product" class="mw-redirect" title="Wedge product">wedge product</a>, which is similar to the <a href="/wiki/Cross_product" title="Cross product">cross product</a> in the sense that the wedge product of two forms representing oriented lengths represents an oriented area. A two-form can be integrated over an oriented surface, and the resulting integral is equivalent to the surface integral giving the flux of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\mathbf {i} +F\mathbf {j} +G\mathbf {k} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>+</mo> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>+</mo> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\mathbf {i} +F\mathbf {j} +G\mathbf {k} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94ab78cf97890001a9a5bf1671669da76dab26f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.993ex; height:2.509ex;" alt="{\displaystyle E\mathbf {i} +F\mathbf {j} +G\mathbf {k} }"></span>. </p><p>Unlike the cross product, and the three-dimensional vector calculus, the wedge product and the calculus of differential forms makes sense in arbitrary dimension and on more general manifolds (curves, surfaces, and their higher-dimensional analogs). The <a href="/wiki/Exterior_derivative" title="Exterior derivative">exterior derivative</a> plays the role of the <a href="/wiki/Gradient" title="Gradient">gradient</a> and <a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)">curl</a> of vector calculus, and <a href="/wiki/Stokes%27_theorem" title="Stokes' theorem">Stokes' theorem</a> simultaneously generalizes the three theorems of vector calculus: the <a href="/wiki/Divergence_theorem" title="Divergence theorem">divergence theorem</a>, <a href="/wiki/Green%27s_theorem" title="Green's theorem">Green's theorem</a>, and the <a href="/wiki/Kelvin-Stokes_theorem" class="mw-redirect" title="Kelvin-Stokes theorem">Kelvin-Stokes theorem</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Summations">Summations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=26" title="Edit section: Summations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Summation#Approximation_by_definite_integrals" title="Summation">Summation § Approximation by definite integrals</a></div> <p>The discrete equivalent of integration is <a href="/wiki/Summation" title="Summation">summation</a>. Summations and integrals can be put on the same foundations using the theory of <a href="/wiki/Lebesgue_integral" title="Lebesgue integral">Lebesgue integrals</a> or <a href="/wiki/Time-scale_calculus" title="Time-scale calculus">time-scale calculus</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Functional_integrals">Functional integrals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=27" title="Edit section: Functional integrals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Functional_integration" title="Functional integration">Functional integration</a></div> <p>An integration that is performed not over a variable (or, in physics, over a space or time dimension), but over a <a href="/wiki/Function_space" title="Function space">space of functions</a>, is referred to as a <a href="/wiki/Functional_integral" class="mw-redirect" title="Functional integral">functional integral</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=28" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Integrals are used extensively in many areas. For example, in <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>, integrals are used to determine the probability of some <a href="/wiki/Random_variable" title="Random variable">random variable</a> falling within a certain range.<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> Moreover, the integral under an entire <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a> must equal 1, which provides a test of whether a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> with no negative values could be a density function or not.<sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> </p><p>Integrals can be used for computing the <a href="/wiki/Area" title="Area">area</a> of a two-dimensional region that has a curved boundary, as well as <a href="/wiki/Volume_integral" title="Volume integral">computing the volume</a> of a three-dimensional object that has a curved boundary. The area of a two-dimensional region can be calculated using the aforementioned definite integral.<sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> The volume of a three-dimensional object such as a disc or washer can be computed by <a href="/wiki/Disc_integration" title="Disc integration">disc integration</a> using the equation for the volume of a cylinder, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi r^{2}h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi r^{2}h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f0350a9c71cfb1388ec50689af098b6be292acd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.774ex; height:2.676ex;" alt="{\displaystyle \pi r^{2}h}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> is the radius. In the case of a simple disc created by rotating a curve about the <span class="texhtml"><i>x</i></span>-axis, the radius is given by <span class="texhtml"><i>f</i>(<i>x</i>)</span>, and its height is the differential <span class="texhtml"><i>dx</i></span>. Using an integral with bounds <span class="texhtml"><i>a</i></span> and <span class="texhtml"><i>b</i></span>, the volume of the disc is equal to:<sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \int _{a}^{b}f^{2}(x)\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \int _{a}^{b}f^{2}(x)\,dx.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/408abd7f1c46b3061c00228796685a6be44558ef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.601ex; height:6.343ex;" alt="{\displaystyle \pi \int _{a}^{b}f^{2}(x)\,dx.}"></span>Integrals are also used in physics, in areas like <a href="/wiki/Kinematics" title="Kinematics">kinematics</a> to find quantities like <a href="/wiki/Displacement_(vector)" class="mw-redirect" title="Displacement (vector)">displacement</a>, <a href="/wiki/Time" title="Time">time</a>, and <a href="/wiki/Velocity" title="Velocity">velocity</a>. For example, in <a href="/wiki/Rectilinear_motion" class="mw-redirect" title="Rectilinear motion">rectilinear motion</a>, the displacement of an object over the time interval <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"></span> is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(b)-x(a)=\int _{a}^{b}v(t)\,dt,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(b)-x(a)=\int _{a}^{b}v(t)\,dt,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7745483341530cfe8966a0469eed80495695256" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.099ex; height:6.343ex;" alt="{\displaystyle x(b)-x(a)=\int _{a}^{b}v(t)\,dt,}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/243a0bf98a12f48552ba6a70302122d81b237b3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.777ex; height:2.843ex;" alt="{\displaystyle v(t)}"></span> is the velocity expressed as a function of time.<sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> The work done by a force <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71a82805d469cdfa7856c11d6ee756acd1dc7174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.88ex; height:2.843ex;" alt="{\displaystyle F(x)}"></span> (given as a function of position) from an initial position <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> to a final position <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> is:<sup id="cite_ref-54" class="reference"><a href="#cite_note-54"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{A\rightarrow B}=\int _{A}^{B}F(x)\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{A\rightarrow B}=\int _{A}^{B}F(x)\,dx.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cef9f21422022e97b692fe1c1d245fc67803d367" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.438ex; height:6.176ex;" alt="{\displaystyle W_{A\rightarrow B}=\int _{A}^{B}F(x)\,dx.}"></span></dd></dl> <p>Integrals are also used in <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a>, where <a href="/wiki/Thermodynamic_integration" title="Thermodynamic integration">thermodynamic integration</a> is used to calculate the difference in free energy between two given states. </p> <div class="mw-heading mw-heading2"><h2 id="Computation">Computation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=29" title="Edit section: Computation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Analytical">Analytical</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=30" title="Edit section: Analytical"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The most basic technique for computing definite integrals of one real variable is based on the <a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">fundamental theorem of calculus</a>. Let <span class="texhtml"><i>f</i>(<i>x</i>)</span> be the function of <span class="texhtml mvar" style="font-style:italic;">x</span> to be integrated over a given interval <span class="texhtml">[<i>a</i>, <i>b</i>]</span>. Then, find an antiderivative of <span class="texhtml mvar" style="font-style:italic;">f</span>; that is, a function <span class="texhtml mvar" style="font-style:italic;">F</span> such that <span class="texhtml"><i>F</i>′ = <i>f</i></span> on the interval. Provided the integrand and integral have no <a href="/wiki/Mathematical_singularity" class="mw-redirect" title="Mathematical singularity">singularities</a> on the path of integration, by the fundamental theorem of calculus, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}f(x)\,dx=F(b)-F(a).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)\,dx=F(b)-F(a).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f470e7743fda04c3d353a4dee2f441ae454f528" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.052ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(x)\,dx=F(b)-F(a).}"></span></dd></dl> <p>Sometimes it is necessary to use one of the many techniques that have been developed to evaluate integrals. Most of these techniques rewrite one integral as a different one which is hopefully more tractable. Techniques include <a href="/wiki/Integration_by_substitution" title="Integration by substitution">integration by substitution</a>, <a href="/wiki/Integration_by_parts" title="Integration by parts">integration by parts</a>, <a href="/wiki/Trigonometric_substitution" title="Trigonometric substitution">integration by trigonometric substitution</a>, and <a href="/wiki/Partial_fractions_in_integration" class="mw-redirect" title="Partial fractions in integration">integration by partial fractions</a>. </p><p>Alternative methods exist to compute more complex integrals. Many <a href="/wiki/Nonelementary_integral" title="Nonelementary integral">nonelementary integrals</a> can be expanded in a <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> and integrated term by term. Occasionally, the resulting infinite series can be summed analytically. The method of convolution using <a href="/wiki/Meijer_G-function" title="Meijer G-function">Meijer G-functions</a> can also be used, assuming that the integrand can be written as a product of Meijer G-functions. There are also many less common ways of calculating definite integrals; for instance, <a href="/wiki/Parseval%27s_identity" title="Parseval's identity">Parseval's identity</a> can be used to transform an integral over a rectangular region into an infinite sum. Occasionally, an integral can be evaluated by a trick; for an example of this, see <a href="/wiki/Gaussian_integral" title="Gaussian integral">Gaussian integral</a>. </p><p>Computations of volumes of <a href="/wiki/Solid_of_revolution" title="Solid of revolution">solids of revolution</a> can usually be done with <a href="/wiki/Disk_integration" class="mw-redirect" title="Disk integration">disk integration</a> or <a href="/wiki/Shell_integration" title="Shell integration">shell integration</a>. </p><p>Specific results which have been worked out by various techniques are collected in the <a href="/wiki/Lists_of_integrals" title="Lists of integrals">list of integrals</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Symbolic">Symbolic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=31" title="Edit section: Symbolic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Symbolic_integration" title="Symbolic integration">Symbolic integration</a></div> <p>Many problems in mathematics, physics, and engineering involve integration where an explicit formula for the integral is desired. Extensive <a href="/wiki/Lists_of_integrals" title="Lists of integrals">tables of integrals</a> have been compiled and published over the years for this purpose. With the spread of computers, many professionals, educators, and students have turned to <a href="/wiki/Computer_algebra_system" title="Computer algebra system">computer algebra systems</a> that are specifically designed to perform difficult or tedious tasks, including integration. Symbolic integration has been one of the motivations for the development of the first such systems, like <a href="/wiki/Macsyma" title="Macsyma">Macsyma</a> and <a href="/wiki/Maple_(software)" title="Maple (software)">Maple</a>. </p><p>A major mathematical difficulty in symbolic integration is that in many cases, a relatively simple function does not have integrals that can be expressed in <a href="/wiki/Closed-form_expression" title="Closed-form expression">closed form</a> involving only <a href="/wiki/Elementary_function" title="Elementary function">elementary functions</a>, include <a href="/wiki/Rational_function" title="Rational function">rational</a> and <a href="/wiki/Exponential_function" title="Exponential function">exponential</a> functions, <a href="/wiki/Logarithm" title="Logarithm">logarithm</a>, <a href="/wiki/Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a> and <a href="/wiki/Inverse_trigonometric_functions" title="Inverse trigonometric functions">inverse trigonometric functions</a>, and the operations of multiplication and composition. The <a href="/wiki/Risch_algorithm" title="Risch algorithm">Risch algorithm</a> provides a general criterion to determine whether the antiderivative of an elementary function is elementary and to compute the integral if is elementary. However, functions with closed expressions of antiderivatives are the exception, and consequently, computerized algebra systems have no hope of being able to find an antiderivative for a randomly constructed elementary function. On the positive side, if the 'building blocks' for antiderivatives are fixed in advance, it may still be possible to decide whether the antiderivative of a given function can be expressed using these blocks and operations of multiplication and composition and to find the symbolic answer whenever it exists. The Risch algorithm, implemented in <a href="/wiki/Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a>, <a href="/wiki/Maple_(software)" title="Maple (software)">Maple</a> and other <a href="/wiki/Computer_algebra_system" title="Computer algebra system">computer algebra systems</a>, does just that for functions and antiderivatives built from rational functions, <a href="/wiki/Nth_root" title="Nth root">radicals</a>, logarithm, and exponential functions. </p><p>Some special integrands occur often enough to warrant special study. In particular, it may be useful to have, in the set of antiderivatives, the <a href="/wiki/Special_functions" title="Special functions">special functions</a> (like the <a href="/wiki/Legendre_function" title="Legendre function">Legendre functions</a>, the <a href="/wiki/Hypergeometric_function" title="Hypergeometric function">hypergeometric function</a>, the <a href="/wiki/Gamma_function" title="Gamma function">gamma function</a>, the <a href="/wiki/Incomplete_gamma_function" title="Incomplete gamma function">incomplete gamma function</a> and so on). Extending Risch's algorithm to include such functions is possible but challenging and has been an active research subject. </p><p>More recently a new approach has emerged, using <a href="/wiki/D-finite_function" class="mw-redirect" title="D-finite function"> <i>D</i>-finite functions</a>, which are the solutions of <a href="/wiki/Linear_differential_equation" title="Linear differential equation">linear differential equations</a> with polynomial coefficients. Most of the elementary and special functions are <i>D</i>-finite, and the integral of a <i>D</i>-finite function is also a <i>D</i>-finite function. This provides an algorithm to express the antiderivative of a <i>D</i>-finite function as the solution of a differential equation. This theory also allows one to compute the definite integral of a <i>D</i>-function as the sum of a series given by the first coefficients and provides an algorithm to compute any coefficient. </p><p>Rule-based integration systems facilitate integration. Rubi, a computer algebra system rule-based integrator, pattern matches an extensive system of symbolic integration rules to integrate a wide variety of integrands. This system uses over 6600 integration rules to compute integrals.<sup id="cite_ref-FOOTNOTERichScheibeAbbasi2018_55-0" class="reference"><a href="#cite_note-FOOTNOTERichScheibeAbbasi2018-55"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Ramanujan%27s_master_theorem#Bracket_integration_method" title="Ramanujan's master theorem"> method of brackets</a> is a generalization of Ramanujan's master theorem that can be applied to a wide range of univariate and multivariate integrals. A set of rules are applied to the coefficients and exponential terms of the integrand's power series expansion to determine the integral. The method is closely related to the <a href="/wiki/Mellin_transform" title="Mellin transform">Mellin transform</a>.<sup id="cite_ref-FOOTNOTEGonzalezJiuMoll2020_56-0" class="reference"><a href="#cite_note-FOOTNOTEGonzalezJiuMoll2020-56"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Numerical">Numerical</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=32" title="Edit section: Numerical"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Numerical_integration" title="Numerical integration">Numerical integration</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Numerical_quadrature_4up.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Numerical_quadrature_4up.png/220px-Numerical_quadrature_4up.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Numerical_quadrature_4up.png/330px-Numerical_quadrature_4up.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Numerical_quadrature_4up.png/440px-Numerical_quadrature_4up.png 2x" data-file-width="1260" data-file-height="1260" /></a><figcaption>Numerical quadrature methods: rectangle method, trapezoidal rule, Romberg's method, Gaussian quadrature</figcaption></figure> <p>Definite integrals may be approximated using several methods of <a href="/wiki/Numerical_integration" title="Numerical integration">numerical integration</a>. The <a href="/wiki/Rectangle_method" class="mw-redirect" title="Rectangle method">rectangle method</a> relies on dividing the region under the function into a series of rectangles corresponding to function values and multiplies by the step width to find the sum. A better approach, the <a href="/wiki/Trapezoidal_rule" title="Trapezoidal rule">trapezoidal rule</a>, replaces the rectangles used in a Riemann sum with trapezoids. The trapezoidal rule weights the first and last values by one half, then multiplies by the step width to obtain a better approximation.<sup id="cite_ref-57" class="reference"><a href="#cite_note-57"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> The idea behind the trapezoidal rule, that more accurate approximations to the function yield better approximations to the integral, can be carried further: <a href="/wiki/Simpson%27s_rule" title="Simpson's rule">Simpson's rule</a> approximates the integrand by a piecewise quadratic function.<sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup> </p><p>Riemann sums, the trapezoidal rule, and Simpson's rule are examples of a family of quadrature rules called the <a href="/wiki/Newton%E2%80%93Cotes_formulas" title="Newton–Cotes formulas">Newton–Cotes formulas</a>. The degree <span class="texhtml mvar" style="font-style:italic;">n</span> Newton–Cotes quadrature rule approximates the polynomial on each subinterval by a degree <i><span class="texhtml mvar" style="font-style:italic;">n</span></i> polynomial. This polynomial is chosen to interpolate the values of the function on the interval.<sup id="cite_ref-59" class="reference"><a href="#cite_note-59"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> Higher degree Newton–Cotes approximations can be more accurate, but they require more function evaluations, and they can suffer from numerical inaccuracy due to <a href="/wiki/Runge%27s_phenomenon" title="Runge's phenomenon">Runge's phenomenon</a>. One solution to this problem is <a href="/wiki/Clenshaw%E2%80%93Curtis_quadrature" title="Clenshaw–Curtis quadrature">Clenshaw–Curtis quadrature</a>, in which the integrand is approximated by expanding it in terms of <a href="/wiki/Chebyshev_polynomials" title="Chebyshev polynomials">Chebyshev polynomials</a>. </p><p><a href="/wiki/Romberg%27s_method" title="Romberg's method">Romberg's method</a> halves the step widths incrementally, giving trapezoid approximations denoted by <span class="texhtml"><i>T</i>(<i>h</i><sub>0</sub>)</span>, <span class="texhtml"><i>T</i>(<i>h</i><sub>1</sub>)</span>, and so on, where <span class="texhtml"><i>h</i><sub><i>k</i>+1</sub></span> is half of <span class="texhtml"><i>h</i><sub><i>k</i></sub></span>. For each new step size, only half the new function values need to be computed; the others carry over from the previous size. It then <a href="/wiki/Interpolation" title="Interpolation">interpolate</a> a polynomial through the approximations, and extrapolate to <span class="texhtml"><i>T</i>(0)</span>. <a href="/wiki/Gaussian_quadrature" title="Gaussian quadrature">Gaussian quadrature</a> evaluates the function at the roots of a set of <a href="/wiki/Orthogonal_polynomials" title="Orthogonal polynomials">orthogonal polynomials</a>.<sup id="cite_ref-60" class="reference"><a href="#cite_note-60"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup> An <span class="texhtml mvar" style="font-style:italic;">n</span>-point Gaussian method is exact for polynomials of degree up to <span class="texhtml">2<i>n</i> − 1</span>. </p><p>The computation of higher-dimensional integrals (for example, volume calculations) makes important use of such alternatives as <a href="/wiki/Monte_Carlo_integration" title="Monte Carlo integration">Monte Carlo integration</a>.<sup id="cite_ref-61" class="reference"><a href="#cite_note-61"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Mechanical">Mechanical</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=33" title="Edit section: Mechanical"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The area of an arbitrary two-dimensional shape can be determined using a measuring instrument called <a href="/wiki/Planimeter" title="Planimeter">planimeter</a>. The volume of irregular objects can be measured with precision by the fluid <a href="/wiki/Displacement_(fluid)" title="Displacement (fluid)">displaced</a> as the object is submerged. </p> <div class="mw-heading mw-heading3"><h3 id="Geometrical">Geometrical</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=34" title="Edit section: Geometrical"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Quadrature_(mathematics)" class="mw-redirect" title="Quadrature (mathematics)">Quadrature (mathematics)</a></div> <p>Area can sometimes be found via <a href="/wiki/Geometrical" class="mw-redirect" title="Geometrical">geometrical</a> <a href="/wiki/Compass-and-straightedge_construction" class="mw-redirect" title="Compass-and-straightedge construction">compass-and-straightedge constructions</a> of an equivalent <a href="/wiki/Square" title="Square">square</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Integration_by_differentiation">Integration by differentiation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=35" title="Edit section: Integration by differentiation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Kempf, Jackson and Morales demonstrated mathematical relations that allow an integral to be calculated by means of <a href="/wiki/Derivative" title="Derivative">differentiation</a>. Their calculus involves the <a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a> and the <a href="/wiki/Partial_derivative" title="Partial derivative">partial derivative</a> operator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57563dabdda89ee216b5a897bf9f78f5c7030eb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.407ex; height:2.509ex;" alt="{\displaystyle \partial _{x}}"></span>. This can also be applied to <a href="/wiki/Functional_integral" class="mw-redirect" title="Functional integral">functional integrals</a>, allowing them to be computed by <a href="/wiki/Functional_derivative" title="Functional derivative">functional differentiation</a>.<sup id="cite_ref-62" class="reference"><a href="#cite_note-62"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=36" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Using_the_fundamental_theorem_of_calculus">Using the fundamental theorem of calculus</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=37" title="Edit section: Using the fundamental theorem of calculus"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">fundamental theorem of calculus</a> allows straightforward calculations of basic functions: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{\pi }\sin(x)\,dx=-\cos(x){\big |}_{x=0}^{x=\pi }=-\cos(\pi )-{\big (}-\cos(0){\big )}=2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>=</mo> <mi>π</mi> </mrow> </msubsup> <mo>=</mo> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\pi }\sin(x)\,dx=-\cos(x){\big |}_{x=0}^{x=\pi }=-\cos(\pi )-{\big (}-\cos(0){\big )}=2.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24d26f0c82508709b27d054bdc0cf5690ef20b53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:58.883ex; height:5.843ex;" alt="{\displaystyle \int _{0}^{\pi }\sin(x)\,dx=-\cos(x){\big |}_{x=0}^{x=\pi }=-\cos(\pi )-{\big (}-\cos(0){\big )}=2.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=38" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1259569809">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{clear:left;float:left;margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><span class="portalbox-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></span><span class="portalbox-link"><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></span></li></ul> <ul><li><a href="/wiki/Integral_equation" title="Integral equation">Integral equation</a> – Equations with an unknown function under an integral sign</li> <li><a href="/wiki/Integral_symbol" title="Integral symbol">Integral symbol</a> – Mathematical symbol used to denote integrals and antiderivatives</li> <li><a href="/wiki/Lists_of_integrals" title="Lists of integrals">Lists of integrals</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=39" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Integral calculus is a very well established mathematical discipline for which there are many sources. See <a href="#CITEREFApostol1967">Apostol 1967</a> and <a href="#CITEREFAntonBivensDavis2016">Anton, Bivens & Davis 2016</a>, for example.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=40" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="#CITEREFBurton2011">Burton 2011</a>, p. 117.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a href="#CITEREFHeath2002">Heath 2002</a>.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a href="#CITEREFKatz2009">Katz 2009</a>, pp. 201–204.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a href="#CITEREFKatz2009">Katz 2009</a>, pp. 284–285.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFDennisKreinovichRump1998" class="citation journal cs1">Dennis, David; Kreinovich, Vladik; Rump, Siegfried M. (1998-05-01). <a rel="nofollow" class="external text" href="https://doi.org/10.1023/A:1009989211143">"Intervals and the Origins of Calculus"</a>. <i>Reliable Computing</i>. <b>4</b> (2): 191–197. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FA%3A1009989211143">10.1023/A:1009989211143</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1573-1340">1573-1340</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Reliable+Computing&rft.atitle=Intervals+and+the+Origins+of+Calculus&rft.volume=4&rft.issue=2&rft.pages=191-197&rft.date=1998-05-01&rft_id=info%3Adoi%2F10.1023%2FA%3A1009989211143&rft.issn=1573-1340&rft.aulast=Dennis&rft.aufirst=David&rft.au=Kreinovich%2C+Vladik&rft.au=Rump%2C+Siegfried+M.&rft_id=https%3A%2F%2Fdoi.org%2F10.1023%2FA%3A1009989211143&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a href="#CITEREFKatz2009">Katz 2009</a>, pp. 305–306.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a href="#CITEREFKatz2009">Katz 2009</a>, pp. 516–517.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><a href="#CITEREFStruik1986">Struik 1986</a>, pp. 215–216.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><a href="#CITEREFKatz2009">Katz 2009</a>, pp. 536–537.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><a href="#CITEREFBurton2011">Burton 2011</a>, pp. 385–386.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><a href="#CITEREFStillwell1989">Stillwell 1989</a>, p. 131.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a href="#CITEREFKatz2009">Katz 2009</a>, pp. 628–629.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><a href="#CITEREFKatz2009">Katz 2009</a>, p. 785.</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><a href="#CITEREFBurton2011">Burton 2011</a>, p. 414; <a href="#CITEREFLeibniz1899">Leibniz 1899</a>, p. 154.</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><a href="#CITEREFCajori1929">Cajori 1929</a>, pp. 249–250; <a href="#CITEREFFourier1822">Fourier 1822</a>, §231.</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><a href="#CITEREFCajori1929">Cajori 1929</a>, p. 246.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><a href="#CITEREFCajori1929">Cajori 1929</a>, p. 182.</span> </li> <li id="cite_note-:1-19"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_19-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_19-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFApostol1967">Apostol 1967</a>, p. 74.</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><a href="#CITEREFAntonBivensDavis2016">Anton, Bivens & Davis 2016</a>, p. 259.</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><a href="#CITEREFApostol1967">Apostol 1967</a>, p. 69.</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><a href="#CITEREFAntonBivensDavis2016">Anton, Bivens & Davis 2016</a>, pp. 286−287.</span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><a href="#CITEREFKrantz1991">Krantz 1991</a>, p. 173.</span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><a href="#CITEREFRudin1987">Rudin 1987</a>, p. 5.</span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><a href="#CITEREFSiegmund-Schultze2008">Siegmund-Schultze 2008</a>, p. 796.</span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><a href="#CITEREFFolland1999">Folland 1999</a>, pp. 57–58.</span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><a href="#CITEREFBourbaki2004">Bourbaki 2004</a>, p. IV.43.</span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><a href="#CITEREFLiebLoss2001">Lieb & Loss 2001</a>, p. 14.</span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><a href="#CITEREFFolland1999">Folland 1999</a>, p. 53.</span> </li> <li id="cite_note-:3-30"><span class="mw-cite-backlink">^ <a href="#cite_ref-:3_30-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:3_30-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFRudin1987">Rudin 1987</a>, p. 25.</span> </li> <li id="cite_note-:0-31"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_31-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_31-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFApostol1967">Apostol 1967</a>, p. 80.</span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text"><a href="#CITEREFRudin1987">Rudin 1987</a>, p. 54.</span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><a href="#CITEREFApostol1967">Apostol 1967</a>, p. 81.</span> </li> <li id="cite_note-:4-34"><span class="mw-cite-backlink">^ <a href="#cite_ref-:4_34-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:4_34-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFRudin1987">Rudin 1987</a>, p. 63.</span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text"><a href="#CITEREFApostol1967">Apostol 1967</a>, p. 202.</span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text"><a href="#CITEREFApostol1967">Apostol 1967</a>, p. 205.</span> </li> <li id="cite_note-FOOTNOTEMontesinosZizlerZizler2015355-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEMontesinosZizlerZizler2015355_37-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFMontesinosZizlerZizler2015">Montesinos, Zizler & Zizler 2015</a>, p. 355.</span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text"><a href="#CITEREFApostol1967">Apostol 1967</a>, p. 416.</span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text"><a href="#CITEREFApostol1967">Apostol 1967</a>, p. 418.</span> </li> <li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text"><a href="#CITEREFAntonBivensDavis2016">Anton, Bivens & Davis 2016</a>, p. 895.</span> </li> <li id="cite_note-:2-41"><span class="mw-cite-backlink">^ <a href="#cite_ref-:2_41-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:2_41-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFAntonBivensDavis2016">Anton, Bivens & Davis 2016</a>, p. 896.</span> </li> <li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text"><a href="#CITEREFAntonBivensDavis2016">Anton, Bivens & Davis 2016</a>, p. 897.</span> </li> <li id="cite_note-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-43">^</a></b></span> <span class="reference-text"><a href="#CITEREFAntonBivensDavis2016">Anton, Bivens & Davis 2016</a>, p. 980.</span> </li> <li id="cite_note-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-44">^</a></b></span> <span class="reference-text"><a href="#CITEREFAntonBivensDavis2016">Anton, Bivens & Davis 2016</a>, p. 981.</span> </li> <li id="cite_note-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-45">^</a></b></span> <span class="reference-text"><a href="#CITEREFAntonBivensDavis2016">Anton, Bivens & Davis 2016</a>, p. 697.</span> </li> <li id="cite_note-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-46">^</a></b></span> <span class="reference-text"><a href="#CITEREFAntonBivensDavis2016">Anton, Bivens & Davis 2016</a>, p. 991.</span> </li> <li id="cite_note-47"><span class="mw-cite-backlink"><b><a href="#cite_ref-47">^</a></b></span> <span class="reference-text"><a href="#CITEREFAntonBivensDavis2016">Anton, Bivens & Davis 2016</a>, p. 1014.</span> </li> <li id="cite_note-48"><span class="mw-cite-backlink"><b><a href="#cite_ref-48">^</a></b></span> <span class="reference-text"><a href="#CITEREFAntonBivensDavis2016">Anton, Bivens & Davis 2016</a>, p. 1024.</span> </li> <li id="cite_note-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-49">^</a></b></span> <span class="reference-text"><a href="#CITEREFFeller1966">Feller 1966</a>, p. 1.</span> </li> <li id="cite_note-50"><span class="mw-cite-backlink"><b><a href="#cite_ref-50">^</a></b></span> <span class="reference-text"><a href="#CITEREFFeller1966">Feller 1966</a>, p. 3.</span> </li> <li id="cite_note-51"><span class="mw-cite-backlink"><b><a href="#cite_ref-51">^</a></b></span> <span class="reference-text"><a href="#CITEREFApostol1967">Apostol 1967</a>, pp. 88–89.</span> </li> <li id="cite_note-52"><span class="mw-cite-backlink"><b><a href="#cite_ref-52">^</a></b></span> <span class="reference-text"><a href="#CITEREFApostol1967">Apostol 1967</a>, pp. 111–114.</span> </li> <li id="cite_note-53"><span class="mw-cite-backlink"><b><a href="#cite_ref-53">^</a></b></span> <span class="reference-text"><a href="#CITEREFAntonBivensDavis2016">Anton, Bivens & Davis 2016</a>, p. 306.</span> </li> <li id="cite_note-54"><span class="mw-cite-backlink"><b><a href="#cite_ref-54">^</a></b></span> <span class="reference-text"><a href="#CITEREFApostol1967">Apostol 1967</a>, p. 116.</span> </li> <li id="cite_note-FOOTNOTERichScheibeAbbasi2018-55"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTERichScheibeAbbasi2018_55-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFRichScheibeAbbasi2018">Rich, Scheibe & Abbasi 2018</a>.</span> </li> <li id="cite_note-FOOTNOTEGonzalezJiuMoll2020-56"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGonzalezJiuMoll2020_56-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGonzalezJiuMoll2020">Gonzalez, Jiu & Moll 2020</a>.</span> </li> <li id="cite_note-57"><span class="mw-cite-backlink"><b><a href="#cite_ref-57">^</a></b></span> <span class="reference-text"><a href="#CITEREFDahlquistBjörck2008">Dahlquist & Björck 2008</a>, pp. 519–520.</span> </li> <li id="cite_note-58"><span class="mw-cite-backlink"><b><a href="#cite_ref-58">^</a></b></span> <span class="reference-text"><a href="#CITEREFDahlquistBjörck2008">Dahlquist & Björck 2008</a>, pp. 522–524.</span> </li> <li id="cite_note-59"><span class="mw-cite-backlink"><b><a href="#cite_ref-59">^</a></b></span> <span class="reference-text"><a href="#CITEREFKahanerMolerNash1989">Kahaner, Moler & Nash 1989</a>, p. 144.</span> </li> <li id="cite_note-60"><span class="mw-cite-backlink"><b><a href="#cite_ref-60">^</a></b></span> <span class="reference-text"><a href="#CITEREFKahanerMolerNash1989">Kahaner, Moler & Nash 1989</a>, p. 147.</span> </li> <li id="cite_note-61"><span class="mw-cite-backlink"><b><a href="#cite_ref-61">^</a></b></span> <span class="reference-text"><a href="#CITEREFKahanerMolerNash1989">Kahaner, Moler & Nash 1989</a>, pp. 139–140.</span> </li> <li id="cite_note-62"><span class="mw-cite-backlink"><b><a href="#cite_ref-62">^</a></b></span> <span class="reference-text"><a href="#CITEREFKempfJacksonMorales2015">Kempf, Jackson & Morales 2015</a>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=41" title="Edit section: Bibliography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 35em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAntonBivensDavis2016" class="citation cs2">Anton, Howard; Bivens, Irl C.; Davis, Stephen (2016), <i>Calculus: Early Transcendentals</i> (11th ed.), John Wiley & Sons, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-118-88382-2" title="Special:BookSources/978-1-118-88382-2"><bdi>978-1-118-88382-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus%3A+Early+Transcendentals&rft.edition=11th&rft.pub=John+Wiley+%26+Sons&rft.date=2016&rft.isbn=978-1-118-88382-2&rft.aulast=Anton&rft.aufirst=Howard&rft.au=Bivens%2C+Irl+C.&rft.au=Davis%2C+Stephen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFApostol1967" class="citation cs2"><a href="/wiki/Tom_M._Apostol" title="Tom M. Apostol">Apostol, Tom M.</a> (1967), <a rel="nofollow" class="external text" href="https://archive.org/details/calculus01apos"><i>Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra</i></a> (2nd ed.), Wiley, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-00005-1" title="Special:BookSources/978-0-471-00005-1"><bdi>978-0-471-00005-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus%2C+Vol.+1%3A+One-Variable+Calculus+with+an+Introduction+to+Linear+Algebra&rft.edition=2nd&rft.pub=Wiley&rft.date=1967&rft.isbn=978-0-471-00005-1&rft.aulast=Apostol&rft.aufirst=Tom+M.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcalculus01apos&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBourbaki2004" class="citation cs2"><a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki, Nicolas</a> (2004), <i>Integration I</i>, Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-41129-1" title="Special:BookSources/3-540-41129-1"><bdi>3-540-41129-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Integration+I&rft.pub=Springer-Verlag&rft.date=2004&rft.isbn=3-540-41129-1&rft.aulast=Bourbaki&rft.aufirst=Nicolas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral" class="Z3988"></span>. In particular chapters III and IV.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBurton2011" class="citation cs2">Burton, David M. (2011), <i>The History of Mathematics: An Introduction</i> (7th ed.), McGraw-Hill, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-338315-6" title="Special:BookSources/978-0-07-338315-6"><bdi>978-0-07-338315-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+History+of+Mathematics%3A+An+Introduction&rft.edition=7th&rft.pub=McGraw-Hill&rft.date=2011&rft.isbn=978-0-07-338315-6&rft.aulast=Burton&rft.aufirst=David+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCajori1929" class="citation cs2"><a href="/wiki/Florian_Cajori" title="Florian Cajori">Cajori, Florian</a> (1929), <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema00cajo_0/page/247"><i>A History Of Mathematical Notations Volume II</i></a>, Open Court Publishing, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-67766-8" title="Special:BookSources/978-0-486-67766-8"><bdi>978-0-486-67766-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+Of+Mathematical+Notations+Volume+II&rft.pub=Open+Court+Publishing&rft.date=1929&rft.isbn=978-0-486-67766-8&rft.aulast=Cajori&rft.aufirst=Florian&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhistoryofmathema00cajo_0%2Fpage%2F247&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDahlquistBjörck2008" class="citation cs2"><a href="/wiki/Germund_Dahlquist" title="Germund Dahlquist">Dahlquist, Germund</a>; Björck, Åke (2008), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070615185623/http://www.mai.liu.se/~akbjo/NMbook.html">"Chapter 5: Numerical Integration"</a>, <i>Numerical Methods in Scientific Computing, Volume I</i>, Philadelphia: <a href="/wiki/Society_for_Industrial_and_Applied_Mathematics" title="Society for Industrial and Applied Mathematics">SIAM</a>, archived from <a rel="nofollow" class="external text" href="http://www.mai.liu.se/~akbjo/NMbook.html">the original</a> on 2007-06-15</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+5%3A+Numerical+Integration&rft.btitle=Numerical+Methods+in+Scientific+Computing%2C+Volume+I&rft.place=Philadelphia&rft.pub=SIAM&rft.date=2008&rft.aulast=Dahlquist&rft.aufirst=Germund&rft.au=Bj%C3%B6rck%2C+%C3%85ke&rft_id=http%3A%2F%2Fwww.mai.liu.se%2F~akbjo%2FNMbook.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeller1966" class="citation cs2"><a href="/wiki/William_Feller" title="William Feller">Feller, William</a> (1966), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontopr02fell_0"><i>An introduction to probability theory and its applications</i></a></span>, John Wiley & Sons</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+probability+theory+and+its+applications&rft.pub=John+Wiley+%26+Sons&rft.date=1966&rft.aulast=Feller&rft.aufirst=William&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontopr02fell_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFolland1999" class="citation cs2"><a href="/wiki/Gerald_Folland" title="Gerald Folland">Folland, Gerald B.</a> (1999), <i>Real Analysis: Modern Techniques and Their Applications</i> (2nd ed.), John Wiley & Sons, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-31716-0" title="Special:BookSources/0-471-31716-0"><bdi>0-471-31716-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Real+Analysis%3A+Modern+Techniques+and+Their+Applications&rft.edition=2nd&rft.pub=John+Wiley+%26+Sons&rft.date=1999&rft.isbn=0-471-31716-0&rft.aulast=Folland&rft.aufirst=Gerald+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFourier1822" class="citation cs2"><a href="/wiki/Joseph_Fourier" title="Joseph Fourier">Fourier, Jean Baptiste Joseph</a> (1822), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=TDQJAAAAIAAJ"><i>Théorie analytique de la chaleur</i></a>, Chez Firmin Didot, père et fils, p. §231</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Th%C3%A9orie+analytique+de+la+chaleur&rft.pages=%C2%A7231&rft.pub=Chez+Firmin+Didot%2C+p%C3%A8re+et+fils&rft.date=1822&rft.aulast=Fourier&rft.aufirst=Jean+Baptiste+Joseph&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DTDQJAAAAIAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral" class="Z3988"></span><br />Available in translation as <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFourier1878" class="citation cs2">Fourier, Joseph (1878), <a rel="nofollow" class="external text" href="https://archive.org/details/analyticaltheory00fourrich"><i>The analytical theory of heat</i></a>, Freeman, Alexander (trans.), Cambridge University Press, pp. 200–201</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+analytical+theory+of+heat&rft.pages=200-201&rft.pub=Cambridge+University+Press&rft.date=1878&rft.aulast=Fourier&rft.aufirst=Joseph&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fanalyticaltheory00fourrich&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGonzalezJiuMoll2020" class="citation cs2">Gonzalez, Ivan; Jiu, Lin; Moll, Victor H. (1 January 2020), "An extension of the method of brackets. Part 2", <i>Open Mathematics</i>, <b>18</b> (1): 983–995, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1707.08942">1707.08942</a></span>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1515%2Fmath-2020-0062">10.1515/math-2020-0062</a></span>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2391-5455">2391-5455</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:222004668">222004668</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Open+Mathematics&rft.atitle=An+extension+of+the+method+of+brackets.+Part+2&rft.volume=18&rft.issue=1&rft.pages=983-995&rft.date=2020-01-01&rft_id=info%3Aarxiv%2F1707.08942&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A222004668%23id-name%3DS2CID&rft.issn=2391-5455&rft_id=info%3Adoi%2F10.1515%2Fmath-2020-0062&rft.aulast=Gonzalez&rft.aufirst=Ivan&rft.au=Jiu%2C+Lin&rft.au=Moll%2C+Victor+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHeath2002" class="citation cs2"><a href="/wiki/Thomas_Little_Heath" class="mw-redirect" title="Thomas Little Heath">Heath, T. L.</a>, ed. (2002), <a rel="nofollow" class="external text" href="https://archive.org/details/worksofarchimede029517mbp"><i>The Works of Archimedes</i></a>, Dover, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-42084-4" title="Special:BookSources/978-0-486-42084-4"><bdi>978-0-486-42084-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Works+of+Archimedes&rft.pub=Dover&rft.date=2002&rft.isbn=978-0-486-42084-4&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fworksofarchimede029517mbp&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral" class="Z3988"></span><br />(Originally published by Cambridge University Press, 1897, based on J. L. Heiberg's Greek version.)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHildebrandt1953" class="citation cs2"><a href="/wiki/Theophil_Henry_Hildebrandt" title="Theophil Henry Hildebrandt">Hildebrandt, T. H.</a> (1953), <a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.bams/1183517761">"Integration in abstract spaces"</a>, <i><a href="/wiki/Bulletin_of_the_American_Mathematical_Society" title="Bulletin of the American Mathematical Society">Bulletin of the American Mathematical Society</a></i>, <b>59</b> (2): 111–139, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9904-1953-09694-X">10.1090/S0002-9904-1953-09694-X</a></span>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0273-0979">0273-0979</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+American+Mathematical+Society&rft.atitle=Integration+in+abstract+spaces&rft.volume=59&rft.issue=2&rft.pages=111-139&rft.date=1953&rft_id=info%3Adoi%2F10.1090%2FS0002-9904-1953-09694-X&rft.issn=0273-0979&rft.aulast=Hildebrandt&rft.aufirst=T.+H.&rft_id=http%3A%2F%2Fprojecteuclid.org%2Feuclid.bams%2F1183517761&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKahanerMolerNash1989" class="citation cs2">Kahaner, David; <a href="/wiki/Cleve_Moler" title="Cleve Moler">Moler, Cleve</a>; Nash, Stephen (1989), "Chapter 5: Numerical Quadrature", <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/numericalmethods0000kaha"><i>Numerical Methods and Software</i></a></span>, Prentice Hall, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-627258-8" title="Special:BookSources/978-0-13-627258-8"><bdi>978-0-13-627258-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+5%3A+Numerical+Quadrature&rft.btitle=Numerical+Methods+and+Software&rft.pub=Prentice+Hall&rft.date=1989&rft.isbn=978-0-13-627258-8&rft.aulast=Kahaner&rft.aufirst=David&rft.au=Moler%2C+Cleve&rft.au=Nash%2C+Stephen&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fnumericalmethods0000kaha&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKallio1966" class="citation cs2">Kallio, Bruce Victor (1966), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140305054035/https://circle.ubc.ca/bitstream/id/132341/UBC_1966_A8%20K3.pdf"><i>A History of the Definite Integral</i></a> <span class="cs1-format">(PDF)</span> (M.A. thesis), University of British Columbia, archived from <a rel="nofollow" class="external text" href="https://open.library.ubc.ca/soa/cIRcle/collections/ubctheses/831/items/1.0080597">the original</a> on 2014-03-05<span class="reference-accessdate">, retrieved <span class="nowrap">2014-02-28</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+the+Definite+Integral&rft.pub=University+of+British+Columbia&rft.date=1966&rft.aulast=Kallio&rft.aufirst=Bruce+Victor&rft_id=https%3A%2F%2Fopen.library.ubc.ca%2Fsoa%2FcIRcle%2Fcollections%2Fubctheses%2F831%2Fitems%2F1.0080597&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKatz2009" class="citation cs2"><a href="/wiki/Victor_J._Katz" title="Victor J. Katz">Katz, Victor J.</a> (2009), <i>A History of Mathematics: An Introduction</i>, <a href="/wiki/Addison-Wesley" title="Addison-Wesley">Addison-Wesley</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-321-38700-4" title="Special:BookSources/978-0-321-38700-4"><bdi>978-0-321-38700-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mathematics%3A+An+Introduction&rft.pub=Addison-Wesley&rft.date=2009&rft.isbn=978-0-321-38700-4&rft.aulast=Katz&rft.aufirst=Victor+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKempfJacksonMorales2015" class="citation cs2">Kempf, Achim; Jackson, David M.; Morales, Alejandro H. (2015), "How to (path-)integrate by differentiating", <i>Journal of Physics: Conference Series</i>, <b>626</b> (1), <a href="/wiki/IOP_Publishing" title="IOP Publishing">IOP Publishing</a>: 012015, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1507.04348">1507.04348</a></span>, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2015JPhCS.626a2015K">2015JPhCS.626a2015K</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1742-6596%2F626%2F1%2F012015">10.1088/1742-6596/626/1/012015</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119642596">119642596</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Physics%3A+Conference+Series&rft.atitle=How+to+%28path-%29integrate+by+differentiating&rft.volume=626&rft.issue=1&rft.pages=012015&rft.date=2015&rft_id=info%3Aarxiv%2F1507.04348&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119642596%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F1742-6596%2F626%2F1%2F012015&rft_id=info%3Abibcode%2F2015JPhCS.626a2015K&rft.aulast=Kempf&rft.aufirst=Achim&rft.au=Jackson%2C+David+M.&rft.au=Morales%2C+Alejandro+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKrantz1991" class="citation cs2"><a href="/wiki/Steven_G._Krantz" title="Steven G. Krantz">Krantz, Steven G.</a> (1991), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=OI-0vu1rb7MC&pg=PA173"><i>Real Analysis and Foundations</i></a>, CRC Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8493-7156-2" title="Special:BookSources/0-8493-7156-2"><bdi>0-8493-7156-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Real+Analysis+and+Foundations&rft.pub=CRC+Press&rft.date=1991&rft.isbn=0-8493-7156-2&rft.aulast=Krantz&rft.aufirst=Steven+G.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DOI-0vu1rb7MC%26pg%3DPA173&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLeibniz1899" class="citation cs2"><a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Leibniz, Gottfried Wilhelm</a> (1899), Gerhardt, Karl Immanuel (ed.), <a rel="nofollow" class="external text" href="http://name.umdl.umich.edu/AAX2762.0001.001"><i>Der Briefwechsel von Gottfried Wilhelm Leibniz mit Mathematikern. Erster Band</i></a>, Berlin: Mayer & Müller</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Der+Briefwechsel+von+Gottfried+Wilhelm+Leibniz+mit+Mathematikern.+Erster+Band&rft.place=Berlin&rft.pub=Mayer+%26+M%C3%BCller&rft.date=1899&rft.aulast=Leibniz&rft.aufirst=Gottfried+Wilhelm&rft_id=http%3A%2F%2Fname.umdl.umich.edu%2FAAX2762.0001.001&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLiebLoss2001" class="citation cs2"><a href="/wiki/Elliott_H._Lieb" title="Elliott H. Lieb">Lieb, Elliott</a>; <a href="/wiki/Michael_Loss" title="Michael Loss">Loss, Michael</a> (2001), <i>Analysis</i>, <a href="/wiki/Graduate_Studies_in_Mathematics" title="Graduate Studies in Mathematics">Graduate Studies in Mathematics</a>, vol. 14 (2nd ed.), <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0821827833" title="Special:BookSources/978-0821827833"><bdi>978-0821827833</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analysis&rft.series=Graduate+Studies+in+Mathematics&rft.edition=2nd&rft.pub=American+Mathematical+Society&rft.date=2001&rft.isbn=978-0821827833&rft.aulast=Lieb&rft.aufirst=Elliott&rft.au=Loss%2C+Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMontesinosZizlerZizler2015" class="citation cs2">Montesinos, Vicente; Zizler, Peter; Zizler, Václav (2015), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mlX1CAAAQBAJ&pg=PA355"><i>An Introduction to Modern Analysis</i></a> (illustrated ed.), Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-12481-0" title="Special:BookSources/978-3-319-12481-0"><bdi>978-3-319-12481-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Modern+Analysis&rft.edition=illustrated&rft.pub=Springer&rft.date=2015&rft.isbn=978-3-319-12481-0&rft.aulast=Montesinos&rft.aufirst=Vicente&rft.au=Zizler%2C+Peter&rft.au=Zizler%2C+V%C3%A1clav&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DmlX1CAAAQBAJ%26pg%3DPA355&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral" class="Z3988"></span></li> <li>Paul J. Nahin (2015), <i>Inside Interesting Integrals</i>, Springer, ISBN 978-1-4939-1276-6.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRichScheibeAbbasi2018" class="citation cs2">Rich, Albert; Scheibe, Patrick; Abbasi, Nasser (16 December 2018), "Rule-based integration: An extensive system of symbolic integration rules", <i>Journal of Open Source Software</i>, <b>3</b> (32): 1073, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2018JOSS....3.1073R">2018JOSS....3.1073R</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.21105%2Fjoss.01073">10.21105/joss.01073</a></span>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:56487062">56487062</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Open+Source+Software&rft.atitle=Rule-based+integration%3A+An+extensive+system+of+symbolic+integration+rules&rft.volume=3&rft.issue=32&rft.pages=1073&rft.date=2018-12-16&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A56487062%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.21105%2Fjoss.01073&rft_id=info%3Abibcode%2F2018JOSS....3.1073R&rft.aulast=Rich&rft.aufirst=Albert&rft.au=Scheibe%2C+Patrick&rft.au=Abbasi%2C+Nasser&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRudin1987" class="citation cs2"><a href="/wiki/Walter_Rudin" title="Walter Rudin">Rudin, Walter</a> (1987), "Chapter 1: Abstract Integration", <i>Real and Complex Analysis</i> (International ed.), McGraw-Hill, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-100276-9" title="Special:BookSources/978-0-07-100276-9"><bdi>978-0-07-100276-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+1%3A+Abstract+Integration&rft.btitle=Real+and+Complex+Analysis&rft.edition=International&rft.pub=McGraw-Hill&rft.date=1987&rft.isbn=978-0-07-100276-9&rft.aulast=Rudin&rft.aufirst=Walter&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSaks1964" class="citation cs2"><a href="/wiki/Stanis%C5%82aw_Saks" title="Stanisław Saks">Saks, Stanisław</a> (1964), <a rel="nofollow" class="external text" href="http://matwbn.icm.edu.pl/kstresc.php?tom=7&wyd=10&jez="><i>Theory of the integral</i></a> (English translation by L. C. Young. With two additional notes by Stefan Banach. Second revised ed.), New York: Dover</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+the+integral&rft.place=New+York&rft.edition=English+translation+by+L.+C.+Young.+With+two+additional+notes+by+Stefan+Banach.+Second+revised&rft.pub=Dover&rft.date=1964&rft.aulast=Saks&rft.aufirst=Stanis%C5%82aw&rft_id=http%3A%2F%2Fmatwbn.icm.edu.pl%2Fkstresc.php%3Ftom%3D7%26wyd%3D10%26jez%3D&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSiegmund-Schultze2008" class="citation cs2"><a href="/wiki/Reinhard_Siegmund-Schultze" title="Reinhard Siegmund-Schultze">Siegmund-Schultze, Reinhard</a> (2008), "Henri Lebesgue", in Timothy Gowers; June Barrow-Green; Imre Leader (eds.), <i>Princeton Companion to Mathematics</i>, Princeton University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-11880-2" title="Special:BookSources/978-0-691-11880-2"><bdi>978-0-691-11880-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Henri+Lebesgue&rft.btitle=Princeton+Companion+to+Mathematics&rft.pub=Princeton+University+Press&rft.date=2008&rft.isbn=978-0-691-11880-2&rft.aulast=Siegmund-Schultze&rft.aufirst=Reinhard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStillwell1989" class="citation cs2"><a href="/wiki/John_Stillwell" title="John Stillwell">Stillwell, John</a> (1989), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/mathematicsitshi0000stil"><i>Mathematics and Its History</i></a></span>, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-96981-0" title="Special:BookSources/0-387-96981-0"><bdi>0-387-96981-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+and+Its+History&rft.pub=Springer&rft.date=1989&rft.isbn=0-387-96981-0&rft.aulast=Stillwell&rft.aufirst=John&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathematicsitshi0000stil&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStoerBulirsch2002" class="citation cs2"><a href="/wiki/Josef_Stoer" title="Josef Stoer">Stoer, Josef</a>; <a href="/wiki/Roland_Bulirsch" title="Roland Bulirsch">Bulirsch, Roland</a> (2002), "Topics in Integration", <i>Introduction to Numerical Analysis</i> (3rd ed.), Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-95452-3" title="Special:BookSources/978-0-387-95452-3"><bdi>978-0-387-95452-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Topics+in+Integration&rft.btitle=Introduction+to+Numerical+Analysis&rft.edition=3rd&rft.pub=Springer&rft.date=2002&rft.isbn=978-0-387-95452-3&rft.aulast=Stoer&rft.aufirst=Josef&rft.au=Bulirsch%2C+Roland&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStruik1986" class="citation cs2"><a href="/wiki/Dirk_Jan_Struik" title="Dirk Jan Struik">Struik, Dirk Jan</a>, ed. (1986), <i>A Source Book in Mathematics, 1200-1800</i>, Princeton, New Jersey: Princeton University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-08404-1" title="Special:BookSources/0-691-08404-1"><bdi>0-691-08404-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Source+Book+in+Mathematics%2C+1200-1800&rft.place=Princeton%2C+New+Jersey&rft.pub=Princeton+University+Press&rft.date=1986&rft.isbn=0-691-08404-1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral" class="Z3988"></span></li> <li>Cornel Ioan Vălean (2019), <i>(Almost Impossible) Integrals, Sums, and Series</i>, Springer, ISBN 978-3-030-02461-1.</li> <li>Cornel Ioan Vălean (2023), <i>More (Almost Impossible) Integrals, Sums, and Series</i>, Springer, ISBN 978-3-031-21261-1.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="http://www.w3.org/TR/arabic-math/">"Arabic mathematical notation"</a>, <i>W3C</i>, 2006</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=W3C&rft.atitle=Arabic+mathematical+notation&rft.date=2006&rft_id=http%3A%2F%2Fwww.w3.org%2FTR%2Farabic-math%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=42" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/40px-Wikibooks-logo-en-noslogan.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/60px-Wikibooks-logo-en-noslogan.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/80px-Wikibooks-logo-en-noslogan.svg.png 2x" data-file-width="400" data-file-height="400" /></span></span></div> <div class="side-box-text plainlist">Wikibooks has a book on the topic of: <i><b><a href="https://en.wikibooks.org/wiki/Calculus" class="extiw" title="wikibooks:Calculus">Calculus</a></b></i></div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Integral">"Integral"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Integral&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DIntegral&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://www.wolframalpha.com/calculators/integral-calculator/">Online Integral Calculator</a>, <a href="/wiki/Wolfram_Alpha" class="mw-redirect" title="Wolfram Alpha">Wolfram Alpha</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Online_books">Online books</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral&action=edit&section=43" title="Edit section: Online books"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Keisler, H. Jerome, <a rel="nofollow" class="external text" href="http://www.math.wisc.edu/~keisler/calc.html">Elementary Calculus: An Approach Using Infinitesimals</a>, University of Wisconsin</li> <li>Stroyan, K. D., <a rel="nofollow" class="external text" href="https://web.archive.org/web/20050911104158/http://www.math.uiowa.edu/~stroyan/InfsmlCalculus/InfsmlCalc.htm">A Brief Introduction to Infinitesimal Calculus</a>, University of Iowa</li> <li>Mauch, Sean, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060415161115/http://www.its.caltech.edu/~sean/book/unabridged.html"><i>Sean's Applied Math Book</i></a>, CIT, an online textbook that includes a complete introduction to calculus</li> <li>Crowell, Benjamin, <a rel="nofollow" class="external text" href="http://www.lightandmatter.com/calc/"><i>Calculus</i></a>, Fullerton College, an online textbook</li> <li>Garrett, Paul, <a rel="nofollow" class="external text" href="http://www.math.umn.edu/~garrett/calculus/">Notes on First-Year Calculus</a></li> <li>Hussain, Faraz, <a rel="nofollow" class="external text" href="http://www.understandingcalculus.com">Understanding Calculus</a>, an online textbook</li> <li>Johnson, William Woolsey (1909) <a rel="nofollow" class="external text" href="http://babel.hathitrust.org/cgi/pt?id=miun.aam9447.0001.001;view=1up;seq=9">Elementary Treatise on Integral Calculus</a>, link from <a href="/wiki/HathiTrust" title="HathiTrust">HathiTrust</a>.</li> <li>Kowalk, W. P., <a rel="nofollow" class="external text" href="http://einstein.informatik.uni-oldenburg.de/20910.html"><i>Integration Theory</i></a>, University of Oldenburg. A new concept to an old problem. Online textbook</li> <li>Sloughter, Dan, <a rel="nofollow" class="external text" href="http://math.furman.edu/~dcs/book">Difference Equations to Differential Equations</a>, an introduction to calculus</li> <li><a rel="nofollow" class="external text" href="http://numericalmethods.eng.usf.edu/topics/integration.html">Numerical Methods of Integration</a> at <i>Holistic Numerical Methods Institute</i></li> <li>P. S. Wang, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060917023831/http://www.lcs.mit.edu/publications/specpub.php?id=660">Evaluation of Definite Integrals by Symbolic Manipulation</a> (1972) — a cookbook of definite integral techniques</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output 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.hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Integrals" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Integrals" title="Template:Integrals"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Integrals" title="Template talk:Integrals"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Integrals" title="Special:EditPage/Template:Integrals"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Integrals" style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">Integrals</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of <br /> integrals</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Riemann_integral" title="Riemann integral">Riemann integral</a></li> <li><a href="/wiki/Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue integral</a></li> <li><a href="/wiki/Burkill_integral" title="Burkill integral">Burkill integral</a></li> <li><a href="/wiki/Bochner_integral" title="Bochner integral">Bochner integral</a></li> <li><a href="/wiki/Daniell_integral" title="Daniell integral">Daniell integral</a></li> <li><a href="/wiki/Darboux_integral" title="Darboux integral">Darboux integral</a></li> <li><a href="/wiki/Henstock%E2%80%93Kurzweil_integral" title="Henstock–Kurzweil integral">Henstock–Kurzweil integral</a></li> <li><a href="/wiki/Haar_measure" title="Haar measure">Haar integral</a></li> <li><a href="/wiki/Hellinger_integral" title="Hellinger integral">Hellinger integral</a></li> <li><a href="/wiki/Khinchin_integral" title="Khinchin integral">Khinchin integral</a></li> <li><a href="/wiki/Kolmogorov_integral" title="Kolmogorov integral">Kolmogorov integral</a></li> <li><a href="/wiki/Lebesgue%E2%80%93Stieltjes_integration" title="Lebesgue–Stieltjes integration">Lebesgue–Stieltjes integral</a></li> <li><a href="/wiki/Pettis_integral" title="Pettis integral">Pettis integral</a></li> <li><a href="/wiki/Pfeffer_integral" title="Pfeffer integral">Pfeffer integral</a></li> <li><a href="/wiki/Riemann%E2%80%93Stieltjes_integral" title="Riemann–Stieltjes integral">Riemann–Stieltjes integral</a></li> <li><a href="/wiki/Regulated_integral" title="Regulated integral">Regulated integral</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Integration <br /> techniques</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Integration_by_substitution" title="Integration by substitution">Substitution</a> <ul><li><a href="/wiki/Trigonometric_substitution" title="Trigonometric substitution">Trigonometric</a></li> <li><a href="/wiki/Euler_substitution" title="Euler substitution">Euler</a></li> <li><a href="/wiki/Weierstrass_substitution" class="mw-redirect" title="Weierstrass substitution">Weierstrass</a></li></ul></li> <li><a href="/wiki/Integration_by_parts" title="Integration by parts">By parts</a></li> <li><a href="/wiki/Integration_by_partial_fractions" class="mw-redirect" title="Integration by partial fractions">Partial fractions</a></li> <li><a href="/wiki/Integration_using_Euler%27s_formula" title="Integration using Euler's formula">Euler's formula</a></li> <li><a href="/wiki/Integral_of_inverse_functions" title="Integral of inverse functions">Inverse functions</a></li> <li><a href="/wiki/Order_of_integration_(calculus)" title="Order of integration (calculus)">Changing order</a></li> <li><a href="/wiki/Integration_by_reduction_formulae" title="Integration by reduction formulae">Reduction formulas</a></li> <li><a href="/wiki/Integration_using_parametric_derivatives" title="Integration using parametric derivatives">Parametric derivatives</a></li> <li><a href="/wiki/Leibniz_integral_rule#Evaluating_definite_integrals" title="Leibniz integral rule">Differentiation under the integral sign</a></li> <li><a href="/wiki/Laplace_transform#Evaluating_improper_integrals" title="Laplace transform">Laplace transform</a></li> <li><a href="/wiki/Contour_integration" title="Contour integration">Contour integration</a></li> <li><a href="/wiki/Laplace%27s_method" title="Laplace's method">Laplace's method</a></li> <li><a href="/wiki/Numerical_integration" title="Numerical integration">Numerical integration</a> <ul><li><a href="/wiki/Simpson%27s_rule" title="Simpson's rule">Simpson's rule</a></li> <li><a href="/wiki/Trapezoidal_rule" title="Trapezoidal rule">Trapezoidal rule</a></li></ul></li> <li><a href="/wiki/Risch_algorithm" title="Risch algorithm">Risch algorithm</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Improper_integral" title="Improper integral">Improper integrals</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Gaussian_integral" title="Gaussian integral">Gaussian integral</a></li> <li><a href="/wiki/Dirichlet_integral" title="Dirichlet integral">Dirichlet integral</a></li> <li>Fermi–Dirac integral <ul><li><a href="/wiki/Complete_Fermi%E2%80%93Dirac_integral" title="Complete Fermi–Dirac integral">complete</a></li> <li><a href="/wiki/Incomplete_Fermi%E2%80%93Dirac_integral" title="Incomplete Fermi–Dirac integral">incomplete</a></li></ul></li> <li><a href="/wiki/Bose%E2%80%93Einstein_integral" class="mw-redirect" title="Bose–Einstein integral">Bose–Einstein integral</a></li> <li><a href="/wiki/Frullani_integral" title="Frullani integral">Frullani integral</a></li> <li><a href="/wiki/Common_integrals_in_quantum_field_theory" title="Common integrals in quantum field theory">Common integrals in quantum field theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Stochastic_integral" class="mw-redirect" title="Stochastic integral">Stochastic integrals</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/It%C3%B4_calculus" title="Itô calculus">Itô integral</a></li> <li><a href="/wiki/Russo%E2%80%93Vallois_integral" title="Russo–Vallois integral">Russo–Vallois integral</a></li> <li><a href="/wiki/Stratonovich_integral" title="Stratonovich integral">Stratonovich integral</a></li> <li><a href="/wiki/Skorokhod_integral" title="Skorokhod integral">Skorokhod integral</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Miscellaneous</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Basel_problem" title="Basel problem">Basel problem</a></li> <li><a href="/wiki/Euler%E2%80%93Maclaurin_formula" title="Euler–Maclaurin formula">Euler–Maclaurin formula</a></li> <li><a href="/wiki/Gabriel%27s_horn" title="Gabriel's horn">Gabriel's horn</a></li> <li><a href="/wiki/Integration_Bee" title="Integration Bee">Integration Bee</a></li> <li><a href="/wiki/Proof_that_22/7_exceeds_%CF%80" title="Proof that 22/7 exceeds π">Proof that 22/7 exceeds π</a></li> <li>Volumes <ul><li><a href="/wiki/Disc_integration" title="Disc integration">Washers</a></li> <li><a href="/wiki/Shell_integration" title="Shell integration">Shells</a></li></ul></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Lp_spaces" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Lp_spaces" title="Template:Lp spaces"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Lp_spaces" title="Template talk:Lp spaces"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Lp_spaces" title="Special:EditPage/Template:Lp spaces"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Lp_spaces" style="font-size:114%;margin:0 4em"><a href="/wiki/Lp_space" title="Lp space">Lp spaces</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_space" title="Banach space">Banach</a> & <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert spaces</a></li> <li><a href="/wiki/Lp_space" title="Lp space"><i>L</i><sup><i>p</i></sup> spaces</a></li> <li><a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">Measure</a> <ul><li><a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue</a></li></ul></li> <li><a href="/wiki/Measure_space" title="Measure space">Measure space</a></li> <li><a href="/wiki/Measurable_space" title="Measurable space">Measurable space</a>/<a href="/wiki/Measurable_function" title="Measurable function">function</a></li> <li><a href="/wiki/Minkowski_distance" title="Minkowski distance">Minkowski distance</a></li> <li><a href="/wiki/Sequence_space" title="Sequence space">Sequence spaces</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/L1_space" class="mw-redirect" title="L1 space"><i>L</i><sup>1</sup> spaces</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Integrable_function" class="mw-redirect" title="Integrable function">Integrable function</a></li> <li><a href="/wiki/Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue integration</a></li> <li><a href="/wiki/Taxicab_geometry" title="Taxicab geometry">Taxicab geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/L2_space" class="mw-redirect" title="L2 space"><i>L</i><sup>2</sup> spaces</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bessel%27s_inequality" title="Bessel's inequality">Bessel's</a></li> <li><a href="/wiki/Cauchy%E2%80%93Schwarz_inequality" title="Cauchy–Schwarz inequality">Cauchy–Schwarz</a></li> <li><a href="/wiki/Euclidean_distance" title="Euclidean distance">Euclidean distance</a></li> <li><a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a></li> <li><a href="/wiki/Parseval%27s_identity" title="Parseval's identity">Parseval's identity</a></li> <li><a href="/wiki/Polarization_identity" title="Polarization identity">Polarization identity</a></li> <li><a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a></li> <li><a href="/wiki/Square-integrable_function" title="Square-integrable function">Square-integrable function</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/L-infinity" title="L-infinity"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9ab400cc4dfd865180cd84c72dc894ca457671f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.458ex; height:2.343ex;" alt="{\displaystyle L^{\infty }}"></span> spaces</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bounded_function" title="Bounded function">Bounded function</a></li> <li><a href="/wiki/Chebyshev_distance" title="Chebyshev distance">Chebyshev distance</a></li> <li><a href="/wiki/Infimum_and_supremum" title="Infimum and supremum">Infimum and supremum</a> <ul><li><a href="/wiki/Essential_infimum_and_essential_supremum" title="Essential infimum and essential supremum">Essential</a></li></ul></li> <li><a href="/wiki/Uniform_norm" title="Uniform norm">Uniform norm</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Maps</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Almost_everywhere" title="Almost everywhere">Almost everywhere</a></li> <li><a href="/wiki/Convergence_almost_everywhere" class="mw-redirect" title="Convergence almost everywhere">Convergence almost everywhere</a></li> <li><a href="/wiki/Convergence_in_measure" title="Convergence in measure">Convergence in measure</a></li> <li><a href="/wiki/Function_space" title="Function space">Function space</a></li> <li><a href="/wiki/Integral_transform" title="Integral transform">Integral transform</a></li> <li><a href="/wiki/Locally_integrable_function" title="Locally integrable function">Locally integrable function</a></li> <li><a href="/wiki/Measurable_function" title="Measurable function">Measurable function</a></li> <li><a href="/wiki/Symmetric_decreasing_rearrangement" title="Symmetric decreasing rearrangement">Symmetric decreasing rearrangement</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Inequalities</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Babenko%E2%80%93Beckner_inequality" title="Babenko–Beckner inequality">Babenko–Beckner</a></li> <li><a href="/wiki/Chebyshev%27s_inequality" title="Chebyshev's inequality">Chebyshev's</a></li> <li><a href="/wiki/Clarkson%27s_inequalities" title="Clarkson's inequalities">Clarkson's</a></li> <li><a href="/wiki/Hanner%27s_inequalities" title="Hanner's inequalities">Hanner's</a></li> <li><a href="/wiki/Hausdorff%E2%80%93Young_inequality" title="Hausdorff–Young inequality">Hausdorff–Young</a></li> <li><a href="/wiki/H%C3%B6lder%27s_inequality" title="Hölder's inequality">Hölder's</a></li> <li><a href="/wiki/Markov%27s_inequality" title="Markov's inequality">Markov's</a></li> <li><a href="/wiki/Minkowski_inequality" title="Minkowski inequality">Minkowski</a></li> <li><a href="/wiki/Young%27s_convolution_inequality" title="Young's convolution inequality">Young's convolution</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Theorems_in_analysis" title="Category:Theorems in analysis">Results</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Marcinkiewicz_interpolation_theorem" title="Marcinkiewicz interpolation theorem">Marcinkiewicz interpolation theorem</a></li> <li><a href="/wiki/Plancherel_theorem" title="Plancherel theorem">Plancherel theorem</a></li> <li><a href="/wiki/Riemann%E2%80%93Lebesgue_lemma" title="Riemann–Lebesgue lemma">Riemann–Lebesgue</a></li> <li><a href="/wiki/Riesz%E2%80%93Fischer_theorem" title="Riesz–Fischer theorem">Riesz–Fischer theorem</a></li> <li><a href="/wiki/Riesz%E2%80%93Thorin_theorem" title="Riesz–Thorin theorem">Riesz–Thorin theorem</a></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><span style="font-size:85%;">For <a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a></span></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Isoperimetric_inequality" title="Isoperimetric inequality">Isoperimetric inequality</a></li> <li><a href="/wiki/Brunn%E2%80%93Minkowski_theorem" title="Brunn–Minkowski theorem">Brunn–Minkowski theorem</a> <ul><li><a href="/wiki/Milman%27s_reverse_Brunn%E2%80%93Minkowski_inequality" title="Milman's reverse Brunn–Minkowski inequality">Milman's reverse</a></li></ul></li> <li><a href="/wiki/Minkowski%E2%80%93Steiner_formula" title="Minkowski–Steiner formula">Minkowski–Steiner formula</a></li> <li><a href="/wiki/Pr%C3%A9kopa%E2%80%93Leindler_inequality" title="Prékopa–Leindler inequality">Prékopa–Leindler inequality</a></li> <li><a href="/wiki/Vitale%27s_random_Brunn%E2%80%93Minkowski_inequality" title="Vitale's random Brunn–Minkowski inequality">Vitale's random Brunn–Minkowski inequality</a></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications & related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bochner_space" title="Bochner space">Bochner space</a></li> <li><a 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error">sMAPE</a></li> <li><a href="/wiki/Mean_absolute_percentage_error" title="Mean absolute percentage error">MAPE</a></li> <li><a href="/wiki/Mean_absolute_scaled_error" title="Mean absolute scaled error">MASE</a></li> <li><a href="/wiki/Mean_squared_prediction_error" title="Mean squared prediction error">MSPE</a></li> <li><a href="/wiki/Root_mean_square" title="Root mean square">RMS</a></li> <li><a href="/wiki/Root-mean-square_deviation" class="mw-redirect" title="Root-mean-square deviation">RMSE/RMSD</a></li> <li><a href="/wiki/Coefficient_of_determination" title="Coefficient of determination">R<sup>2</sup></a></li> <li><a href="/wiki/Mean_directional_accuracy" title="Mean directional accuracy">MDA</a></li> <li><a href="/wiki/Median_absolute_deviation" title="Median absolute deviation">MAD</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Statistical_classification" title="Statistical classification">Classification</a></th><td 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specificity</a></li> <li><a href="/wiki/Cross-entropy#Cross-entropy_loss_function_and_logistic_regression" title="Cross-entropy">Logarithmic Loss</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Cluster_analysis" title="Cluster analysis">Clustering</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Silhouette_(clustering)" title="Silhouette (clustering)">Silhouette</a></li> <li><a href="/wiki/Calinski-Harabasz_index" class="mw-redirect" title="Calinski-Harabasz index">Calinski-Harabasz index</a></li> <li><a href="/wiki/Davies%E2%80%93Bouldin_index" title="Davies–Bouldin index">Davies-Bouldin</a></li> <li><a href="/wiki/Dunn_index" title="Dunn index">Dunn index</a></li> <li><a href="/wiki/Hopkins_statistic" title="Hopkins statistic">Hopkins statistic</a></li> <li><a href="/wiki/Jaccard_index" title="Jaccard index">Jaccard index</a></li> <li><a href="/wiki/Rand_index" title="Rand index">Rand index</a></li> <li><a href="/wiki/Similarity_measure" title="Similarity measure">Similarity measure</a></li> <li><a href="/wiki/Simple_matching_coefficient" title="Simple matching coefficient">SMC</a></li> <li><a href="/wiki/SimHash" title="SimHash">SimHash</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Ranking_(information_retrieval)" title="Ranking (information retrieval)">Ranking</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mean_reciprocal_rank" title="Mean reciprocal rank">MRR</a></li> <li><a href="/wiki/NDCG" class="mw-redirect" title="NDCG">NDCG</a></li> <li><a href="/wiki/Average_precision" class="mw-redirect" title="Average precision">AP</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computer_Vision" class="mw-redirect" title="Computer Vision">Computer Vision</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/PSNR" class="mw-redirect" title="PSNR">PSNR</a></li> <li><a href="/wiki/SSIM" class="mw-redirect" title="SSIM">SSIM</a></li> <li><a href="/wiki/Intersection_over_union" class="mw-redirect" title="Intersection over union">IoU</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Natural_language_processing" title="Natural language processing">NLP</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Perplexity" title="Perplexity">Perplexity</a></li> <li><a href="/wiki/BLEU" title="BLEU">BLEU</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Deep Learning Related Metrics</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Inception_score" title="Inception score">Inception score</a></li> <li><a href="/wiki/Fr%C3%A9chet_inception_distance" title="Fréchet inception distance">FID</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Recommender_system" title="Recommender system">Recommender system</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Coverage_probability" title="Coverage probability">Coverage</a></li> <li><a href="/w/index.php?title=Intra-list_Similarity&action=edit&redlink=1" class="new" title="Intra-list Similarity (page does not exist)">Intra-list Similarity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Similarity</th><td 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