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class="vector-toc-numb">2.2</span> <span>Extensions of the symbol</span> </div> </a> <ul id="toc-Extensions_of_the_symbol-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Typography_in_other_languages" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Typography_in_other_languages"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Typography in other languages</span> </div> </a> <ul id="toc-Typography_in_other_languages-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</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 vector-toc-list-item-expanded"> <a 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</div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Integral symbol</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. 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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_%E0%A6%AA%E0%A7%8D%E0%A6%B0%E0%A6%A4%E0%A7%80%E0%A6%95" title="যোগজীকরণ প্রতীক – Bangla" lang="bn" hreflang="bn" data-title="যোগজীকরণ প্রতীক" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</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_%D0%BF%D0%B0%D0%BB%D0%BB%D0%B8" title="Интеграл палли – Chuvash" lang="cv" hreflang="cv" data-title="Интеграл палли" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Integralzeichen" title="Integralzeichen – German" lang="de" hreflang="de" data-title="Integralzeichen" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Signo_de_integraci%C3%B3n" title="Signo de integración – Spanish" lang="es" hreflang="es" data-title="Signo de integración" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Integralaren_sinbolo" title="Integralaren sinbolo – Basque" lang="eu" hreflang="eu" data-title="Integralaren sinbolo" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</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%EA%B8%B0%ED%98%B8" title="적분기호 – Korean" lang="ko" hreflang="ko" data-title="적분기호" 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%D5%AB_%D5%B6%D5%B7%D5%A1%D5%B6" title="Ինտեգրալի նշան – Armenian" lang="hy" hreflang="hy" data-title="Ինտեգրալի նշան" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/%E2%88%AB" title="∫ – Lithuanian" lang="lt" hreflang="lt" data-title="∫" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%A9%8D%E5%88%86%E8%A8%98%E5%8F%B7" title="積分記号 – Japanese" lang="ja" hreflang="ja" data-title="積分記号" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/S%C3%ADmbolo_de_integral" title="Símbolo de integral – Portuguese" lang="pt" hreflang="pt" data-title="Símbolo de integral" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%97%D0%BD%D0%B0%D0%BA_%D0%B8%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB%D0%B0" title="Знак интеграла – Russian" lang="ru" hreflang="ru" data-title="Знак интеграла" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Simbol_integrala" title="Simbol integrala – Serbian" lang="sr" hreflang="sr" data-title="Simbol integrala" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%97%D0%BD%D0%B0%D0%BA_%D1%96%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB%D0%B0" title="Знак інтеграла – Ukrainian" lang="uk" hreflang="uk" data-title="Знак інтеграла" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%A7%AF%E5%88%86%E7%AC%A6%E5%8F%B7" title="积分符号 – Chinese" lang="zh" hreflang="zh" data-title="积分符号" data-language-autonym="中文" data-language-local-name="Chinese" 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<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=%E2%8E%AE&redirect=no" class="mw-redirect" title="⎮">⎮</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">Mathematical symbol used to denote integrals and antiderivatives</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">Not to be confused with <a href="/wiki/Long_s" title="Long s">Long s</a>, <a href="/wiki/Esh_(letter)" title="Esh (letter)">Esh (letter)</a>, or <a href="/wiki/Voiceless_postalveolar_fricative" title="Voiceless postalveolar fricative">Voiceless postalveolar fricative</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox" style="max-width:20em;"><tbody><tr><th colspan="2" class="infobox-above" style="background: #ccc; font-size: 500%; line-height:1.3; font-weight:normal; font-family:'times new roman', times, georgia, serif, sans-serif;">∫</th></tr><tr><th colspan="2" class="infobox-header" style="background: #ccc;"><div style="background: #ccc; font-size: 125%; font-weight: bold; padding: 0.1em 0.25em 0.2em; line-height: 1.2em;">Integral symbol</div></th></tr><tr><th scope="row" class="infobox-label">In <a href="/wiki/Unicode" title="Unicode">Unicode</a></th><td class="infobox-data"><span class="nowrap"><style data-mw-deduplicate="TemplateStyles:r886049734">.mw-parser-output .monospaced{font-family:monospace,monospace}</style><span class="monospaced">U+222B</span> </span><span style="font-size:125%;line-height:1em">∫</span> <span style="font-variant: small-caps; text-transform: lowercase;">INTEGRAL</span> (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">&int;, &Integral;</span>)</td></tr><tr><th colspan="2" class="infobox-header" style="background: #ccc;">Graphical variants</th></tr><tr><td colspan="2" class="infobox-full-data"><span style="font-size:150%;"><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 \displaystyle \int }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle \int }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e594c389846abbae8442b2c57a4dd70533071047" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:2.194ex; height:5.676ex;" alt="{\displaystyle \displaystyle \int }"></span></span></td></tr><tr><th colspan="2" class="infobox-header" style="background: #ccc;">Different from</th></tr><tr><th scope="row" class="infobox-label">Different from</th><td class="infobox-data"><span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+017F</span> </span><span style="font-size:125%;line-height:1em">ſ</span> <span style="font-variant: small-caps; text-transform: lowercase;"><a href="/wiki/Long_s" title="Long s">LATIN SMALL LETTER LONG S</a></span><br /><span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+0283</span> </span><span style="font-size:125%;line-height:1em">ʃ</span> <span style="font-variant: small-caps; text-transform: lowercase;"><a href="/wiki/Esh_(letter)" title="Esh (letter)">LATIN SMALL LETTER ESH</a></span></td></tr></tbody></table> <p>The <b>integral symbol</b>: </p> <style data-mw-deduplicate="TemplateStyles:r996643573">.mw-parser-output .block-indent{padding-left:3em;padding-right:0;overflow:hidden}</style><div class="block-indent" style="padding-left: 1.5em;"><span style="font-size:200%">∫</span> (<a href="/wiki/Unicode" title="Unicode">Unicode</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 \displaystyle \int }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle \int }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e594c389846abbae8442b2c57a4dd70533071047" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:2.194ex; height:5.676ex;" alt="{\displaystyle \displaystyle \int }"></span> (<a href="/wiki/LaTeX" title="LaTeX">LaTeX</a>)</div> <p>is used to denote <a href="/wiki/Integral" title="Integral">integrals</a> and <a href="/wiki/Antiderivative" title="Antiderivative">antiderivatives</a> in <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, especially in <a href="/wiki/Calculus" title="Calculus">calculus</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_symbol&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">Main article: <a href="/wiki/Leibniz%27s_notation" title="Leibniz's notation">Leibniz's notation</a></div> <p>The notation was introduced by the <a href="/wiki/Germany" title="Germany">German</a> mathematician <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a> in 1675 in his private writings;<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> it first appeared publicly in the article "<span title="Latin-language text"><i lang="la">De Geometria Recondita et analysi indivisibilium atque infinitorum</i></span>" (On a hidden geometry and analysis of indivisibles and infinites), published in <i><a href="/wiki/Acta_Eruditorum" title="Acta Eruditorum">Acta Eruditorum</a></i> in June 1686.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> The symbol was based on the ſ (<a href="/wiki/Long_s" title="Long s">long s</a>) character and was chosen because Leibniz thought of the integral as an infinite <a href="/wiki/Summation" title="Summation">sum</a> of infinitesimal <a href="/wiki/Summand" class="mw-redirect" title="Summand">summands</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Typography_in_Unicode_and_LaTeX">Typography in Unicode and LaTeX</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral_symbol&action=edit&section=2" title="Edit section: Typography in Unicode and LaTeX"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Fundamental_symbol">Fundamental symbol</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral_symbol&action=edit&section=3" title="Edit section: Fundamental symbol"><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/Integral_calculus" class="mw-redirect" title="Integral calculus">Integral calculus</a></div> <p>The integral symbol is <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+222B</span> </span><span style="font-size:125%;line-height:1em">∫</span> <span style="font-variant: small-caps; text-transform: lowercase;">INTEGRAL</span> in <a href="/wiki/Unicode" title="Unicode">Unicode</a><sup id="cite_ref-Unicode_Mathematical_Operators_5-0" class="reference"><a href="#cite_note-Unicode_Mathematical_Operators-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> and <code>\int</code> in <a href="/wiki/LaTeX" title="LaTeX">LaTeX</a>. In <a href="/wiki/HTML" title="HTML">HTML</a>, it is written as <code>&#x222b;</code> (<a href="/wiki/Hexadecimal" title="Hexadecimal">hexadecimal</a>), <code>&#8747;</code> (<a href="/wiki/Decimal" title="Decimal">decimal</a>) and <code>&int;</code> (<a href="/wiki/List_of_XML_and_HTML_character_entity_references#Character_entity_references_in_HTML" title="List of XML and HTML character entity references">named entity</a>). </p><p>The original <a href="/wiki/IBM_PC" class="mw-redirect" title="IBM PC">IBM PC</a> <a href="/wiki/Code_page_437" title="Code page 437">code page 437</a> character set included a couple of characters ⌠ and ⌡ (codes 244 and 245 respectively) to build the integral symbol. These were deprecated in subsequent <a href="/wiki/MS-DOS" title="MS-DOS">MS-DOS</a> <a href="/wiki/Code_page" title="Code page">code pages</a>, but they still remain in <a href="/wiki/Unicode" title="Unicode">Unicode</a> (<a href="/wiki/Miscellaneous_Technical#(2300–233F)" title="Miscellaneous Technical">U+2320 and U+2321</a> respectively) for compatibility. </p><p>The ∫ symbol is very similar to, but not to be confused with, the letter <a href="/wiki/Esh_(letter)" title="Esh (letter)">ʃ</a> ("<a href="/wiki/Esh_(letter)" title="Esh (letter)">esh</a>"). </p> <div class="mw-heading mw-heading3"><h3 id="Extensions_of_the_symbol">Extensions of the symbol</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral_symbol&action=edit&section=4" title="Edit section: Extensions of the symbol"><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/Multiple_integral" title="Multiple integral">Multiple integral</a></div> <p>Related symbols include:<sup id="cite_ref-Unicode_Mathematical_Operators_5-1" class="reference"><a href="#cite_note-Unicode_Mathematical_Operators-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Unicode_Supplemental_Mathematical_Operators_6-0" class="reference"><a href="#cite_note-Unicode_Supplemental_Mathematical_Operators-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <table class="wikitable"> <tbody><tr> <th>Meaning </th> <th scope="col" colspan="2"><a href="/wiki/Unicode" title="Unicode">Unicode</a> </th> <th scope="col" colspan="2"><a href="/wiki/LaTeX" title="LaTeX">LaTeX</a> </th></tr> <tr> <th><a href="/wiki/Double_integral" class="mw-redirect" title="Double integral">Double integral</a> </th> <td style="font-size:20pt;">∬ </td> <td>U+222C </td> <td><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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∬</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iint }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1534cac573e0e2b359c385293720bfca1193dde3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:3.42ex; height:5.676ex;" alt="{\displaystyle \iint }"></span> </td> <td><code>\iint</code> </td></tr> <tr> <th><a href="/wiki/Triple_integral" class="mw-redirect" title="Triple integral">Triple integral</a> </th> <td style="font-size:20pt;">∭ </td> <td>U+222D </td> <td><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 \iiint }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∭</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iiint }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db210c35ec1f4da3138b3ce9549a3e2810868e90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:4.6ex; height:5.676ex;" alt="{\displaystyle \iiint }"></span> </td> <td><code>\iiint</code> </td></tr> <tr> <th><a href="/wiki/Quadruple_integral" class="mw-redirect" title="Quadruple integral">Quadruple integral</a> </th> <td style="font-size:20pt;">⨌ </td> <td>U+2A0C </td> <td><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 \iiiint }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⨌</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iiiint }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ca75d455f98b3334b52b5adb78a0280cd01e596" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:5.939ex; height:5.676ex;" alt="{\displaystyle \iiiint }"></span> </td> <td><code>\iiiint</code> </td></tr> <tr> <th><a href="/wiki/Contour_integral" class="mw-redirect" title="Contour integral">Contour integral</a> </th> <td style="font-size:20pt;">∮ </td> <td>U+222E </td> <td><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 \oint }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∮</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbbea2707a4ee041ddc4da8dd11924d25aa59c89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:2.194ex; height:5.676ex;" alt="{\displaystyle \oint }"></span> </td> <td><code>\oint</code> </td></tr> <tr> <th><a href="/wiki/Clockwise" title="Clockwise">Clockwise</a> integral </th> <td style="font-size:20pt;">∱ </td> <td>U+2231 </td> <td> </td> <td> </td></tr> <tr> <th><a href="/wiki/Counterclockwise" class="mw-redirect" title="Counterclockwise">Counterclockwise</a> integral </th> <td style="font-size:20pt;">⨑ </td> <td>U+2A11 </td> <td> </td> <td> </td></tr> <tr> <th>Clockwise contour integral </th> <td style="font-size:20pt;">∲ </td> <td>U+2232 </td> <td><span class="nowrap mw-no-invert"> <span class="mw-default-size" typeof="mw:File"><span><img alt="\varointclockwise" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/VarointclockwiseLaTeX.svg/17px-VarointclockwiseLaTeX.svg.png" decoding="async" width="17" height="44" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/VarointclockwiseLaTeX.svg/26px-VarointclockwiseLaTeX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/30/VarointclockwiseLaTeX.svg/35px-VarointclockwiseLaTeX.svg.png 2x" data-file-width="141" data-file-height="355" /></span></span><span style="position:relative; right:8px; top:18px; margin-right:-8px;"></span> </span> </td> <td><code>\varointclockwise</code> </td></tr> <tr> <th>Counterclockwise contour integral </th> <td style="font-size:20pt;">∳ </td> <td>U+2233 </td> <td><span class="nowrap mw-no-invert"> <span class="mw-default-size" typeof="mw:File"><span><img alt="\ointctrclockwise" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/OintctrclockwiseLaTeX.svg/17px-OintctrclockwiseLaTeX.svg.png" decoding="async" width="17" height="44" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/OintctrclockwiseLaTeX.svg/26px-OintctrclockwiseLaTeX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bd/OintctrclockwiseLaTeX.svg/35px-OintctrclockwiseLaTeX.svg.png 2x" data-file-width="141" data-file-height="355" /></span></span><span style="position:relative; right:8px; top:18px; margin-right:-8px;"></span> </span> </td> <td><code>\ointctrclockwise</code> </td></tr> <tr> <th><a href="/wiki/Closed_manifold" title="Closed manifold">Closed</a> <a href="/wiki/Surface_integral" title="Surface integral">surface integral</a> </th> <td style="font-size:20pt;">∯ </td> <td>U+222F </td> <td><span class="nowrap mw-no-invert"> <span class="mw-default-size" typeof="mw:File"><span><img alt="\oiint" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/OiintLaTeX.svg/25px-OiintLaTeX.svg.png" decoding="async" width="25" height="44" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/OiintLaTeX.svg/38px-OiintLaTeX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/86/OiintLaTeX.svg/50px-OiintLaTeX.svg.png 2x" data-file-width="204" data-file-height="354" /></span></span><span style="position:relative; right:8px; top:18px; margin-right:-8px;"></span> </span> </td> <td><code>\oiint</code> </td></tr> <tr> <th>Closed <a href="/wiki/Volume_integral" title="Volume integral">volume integral</a> </th> <td style="font-size:20pt;">∰ </td> <td>U+2230 </td> <td><span class="nowrap mw-no-invert"> <span class="mw-default-size" typeof="mw:File"><span><img alt="\oiiint" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/OiiintLaTeX.svg/33px-OiiintLaTeX.svg.png" decoding="async" width="33" height="44" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/OiiintLaTeX.svg/49px-OiiintLaTeX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/OiiintLaTeX.svg/65px-OiiintLaTeX.svg.png 2x" data-file-width="263" data-file-height="353" /></span></span><span style="position:relative; right:8px; top:18px; margin-right:-8px;"></span> </span> </td> <td><code>\oiiint</code> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Typography_in_other_languages">Typography in other languages</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral_symbol&action=edit&section=5" title="Edit section: Typography in other languages"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Integral_Uprightness.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Integral_Uprightness.svg/150px-Integral_Uprightness.svg.png" decoding="async" width="150" height="103" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Integral_Uprightness.svg/225px-Integral_Uprightness.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/71/Integral_Uprightness.svg/300px-Integral_Uprightness.svg.png 2x" data-file-width="162" data-file-height="111" /></a><figcaption>Regional variations (<a href="/wiki/English_language" title="English language">English</a>, <a href="/wiki/German_language" title="German language">German</a>, and <a href="/wiki/Russian_language" title="Russian language">Russian</a> from left to right) of the integral symbol</figcaption></figure> <p>In other languages, the shape of the integral symbol differs slightly from the shape commonly seen in English-language textbooks. While the <a href="/wiki/English_language" title="English language">English</a> integral symbol leans to the right, the <a href="/wiki/German_language" title="German language">German</a> symbol (used throughout <a href="/wiki/Central_Europe" title="Central Europe">Central Europe</a>) is upright, and the <a href="/wiki/Russian_language" title="Russian language">Russian</a> variant leans slightly to the left to occupy less horizontal space.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>Another difference is in the placement of <a href="/wiki/Limits_of_integration" title="Limits of integration">limits</a> for <a href="/wiki/Definite_integral" class="mw-redirect" title="Definite integral">definite integrals</a>. Generally, in English-language books, limits go to the right of the integral symbol: </p><p><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 _{0}^{5}f(t)\,\mathrm {d} t,\quad \int _{g(t)=a}^{g(t)=b}f(t)\,\mathrm {d} t.}"> <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>5</mn> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>,</mo> <mspace width="1em" /> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{5}f(t)\,\mathrm {d} t,\quad \int _{g(t)=a}^{g(t)=b}f(t)\,\mathrm {d} t.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64bfd0666bfcdea1dfa7e36aefe8148c724b825a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.532ex; height:6.676ex;" alt="{\displaystyle \int _{0}^{5}f(t)\,\mathrm {d} t,\quad \int _{g(t)=a}^{g(t)=b}f(t)\,\mathrm {d} t.}"></span> </p><p>By contrast, in German and Russian texts, the limits are placed above and below the integral symbol, and, as a result, the notation requires larger line spacing but is more compact horizontally, especially when using longer expressions in the limits: </p><p><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 \limits _{0}^{T}f(t)\,\mathrm {d} t,\quad \int \limits _{\!\!\!\!\!g(t)=a\!\!\!\!\!}^{\!\!\!\!\!g(t)=b\!\!\!\!\!}f(t)\,\mathrm {d} t.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>,</mo> <mspace width="1em" /> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>b</mi> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \limits _{0}^{T}f(t)\,\mathrm {d} t,\quad \int \limits _{\!\!\!\!\!g(t)=a\!\!\!\!\!}^{\!\!\!\!\!g(t)=b\!\!\!\!\!}f(t)\,\mathrm {d} t.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56367b989d9bfa6f1b5ea9497a3656a7d5bb3d61" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:22.126ex; height:10.009ex;" alt="{\displaystyle \int \limits _{0}^{T}f(t)\,\mathrm {d} t,\quad \int \limits _{\!\!\!\!\!g(t)=a\!\!\!\!\!}^{\!\!\!\!\!g(t)=b\!\!\!\!\!}f(t)\,\mathrm {d} t.}"></span> </p> <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_symbol&action=edit&section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Capital_sigma_notation" class="mw-redirect" title="Capital sigma notation">Capital sigma notation</a></li> <li><a href="/wiki/Capital_pi_notation" class="mw-redirect" title="Capital pi notation">Capital pi notation</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_symbol&action=edit&section=7" 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"> <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">Gottfried Wilhelm Leibniz, <i>Sämtliche Schriften und Briefe, Reihe VII: Mathematische Schriften, vol. 5: Infinitesimalmathematik 1674–1676</i>, Berlin: Akademie Verlag, 2008, pp. <a rel="nofollow" class="external text" href="http://www.gwlb.de/Leibniz/Leibnizarchiv/Veroeffentlichungen/VII5A.pdf">288–295</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20211009052830/https://www.gwlb.de/Leibniz/Leibnizarchiv/Veroeffentlichungen/VII5A.pdf">Archived</a> 2021-10-09 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> ("Analyseos tetragonisticae pars secunda", October 29, 1675) and <a rel="nofollow" class="external text" href="http://www.gwlb.de/Leibniz/Leibnizarchiv/Veroeffentlichungen/VII5B.pdf">321–331</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161003130340/http://www.gwlb.de/Leibniz/Leibnizarchiv/Veroeffentlichungen/VII5B.pdf">Archived</a> 2016-10-03 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> ("Methodi tangentium inversae exempla", November 11, 1675).</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</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="CITEREFAldrich,_John" class="citation web cs1">Aldrich, John. <a rel="nofollow" class="external text" href="http://jeff560.tripod.com/calculus.html">"Earliest Uses of Symbols of Calculus"</a><span class="reference-accessdate">. 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Springer. p. <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicsitshi0000stil/page/110">110</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.pages=110&rft.pub=Springer&rft.date=1989&rft.aulast=Stillwell&rft.aufirst=John&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathematicsitshi0000stil&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral+symbol" class="Z3988"></span></span> </li> <li id="cite_note-Unicode_Mathematical_Operators-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-Unicode_Mathematical_Operators_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Unicode_Mathematical_Operators_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.unicode.org/charts/PDF/U2200.pdf">"Mathematical Operators – Unicode"</a> <span class="cs1-format">(PDF)</span><span class="reference-accessdate">. Retrieved <span class="nowrap">2013-04-26</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Mathematical+Operators+%E2%80%93+Unicode&rft_id=https%3A%2F%2Fwww.unicode.org%2Fcharts%2FPDF%2FU2200.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral+symbol" class="Z3988"></span></span> </li> <li id="cite_note-Unicode_Supplemental_Mathematical_Operators-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Unicode_Supplemental_Mathematical_Operators_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.unicode.org/charts/PDF/U2A00.pdf">"Supplemental Mathematical Operators – Unicode"</a> <span class="cs1-format">(PDF)</span><span class="reference-accessdate">. Retrieved <span class="nowrap">2013-05-05</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Supplemental+Mathematical+Operators+%E2%80%93+Unicode&rft_id=https%3A%2F%2Fwww.unicode.org%2Fcharts%2FPDF%2FU2A00.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral+symbol" 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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120928083018/http://www.uni-giessen.de/partosch/eurotex99/zaitsev.pdf">"Russian Typographical Traditions in Mathematical Literature"</a> <span class="cs1-format">(PDF)</span>. giftbot.toolforge.org. Archived from <a class="external text" href="https://giftbot.toolforge.org/deref.fcgi?url=http%3A%2F%2Fwww.uni-giessen.de%2Fpartosch%2Feurotex99%2Fzaitsev.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 28 September 2012<span class="reference-accessdate">. Retrieved <span class="nowrap">11 October</span> 2021</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Russian+Typographical+Traditions+in+Mathematical+Literature&rft.pub=giftbot.toolforge.org&rft_id=https%3A%2F%2Fgiftbot.toolforge.org%2Fderef.fcgi%3Furl%3Dhttp%253A%252F%252Fwww.uni-giessen.de%252Fpartosch%252Feurotex99%252Fzaitsev.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral+symbol" class="Z3988"></span></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_symbol&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStewart2003" class="citation book cs1">Stewart, James (2003). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/singlevariableca00stew/page/381">"Integrals"</a></span>. <i>Single Variable Calculus: Early Transcendentals</i> (5th ed.). Belmont, CA: Brooks/Cole. p. <a rel="nofollow" class="external text" href="https://archive.org/details/singlevariableca00stew/page/381">381</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-534-39330-6" title="Special:BookSources/0-534-39330-6"><bdi>0-534-39330-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Integrals&rft.btitle=Single+Variable+Calculus%3A+Early+Transcendentals&rft.place=Belmont%2C+CA&rft.pages=381&rft.edition=5th&rft.pub=Brooks%2FCole&rft.date=2003&rft.isbn=0-534-39330-6&rft.aulast=Stewart&rft.aufirst=James&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsinglevariableca00stew%2Fpage%2F381&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral+symbol" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZaitcevJanishewskyBerdnikov1999" class="citation conference cs1">Zaitcev, V.; Janishewsky, A.; Berdnikov, A. (20 September 1999). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170224213730/https://www.staff.uni-giessen.de/~g029/eurotex99/zaitsev.pdf"><i>Russian Typographical Traditions in Mathematical Literature</i></a> <span class="cs1-format">(PDF)</span>. EuroTeX'99. <i>EuroTeX'99 Proceedings</i>. Heidelberg. Archived from <a rel="nofollow" class="external text" href="https://www.staff.uni-giessen.de/~g029/eurotex99/zaitsev.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 24 February 2017.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=conference&rft.jtitle=EuroTeX%2799+Proceedings&rft.atitle=Russian+Typographical+Traditions+in+Mathematical+Literature&rft.date=1999-09-20&rft.aulast=Zaitcev&rft.aufirst=V.&rft.au=Janishewsky%2C+A.&rft.au=Berdnikov%2C+A.&rft_id=https%3A%2F%2Fwww.staff.uni-giessen.de%2F~g029%2Feurotex99%2Fzaitsev.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral+symbol" class="Z3988"></span></li></ul> <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_symbol&action=edit&section=9" title="Edit section: External links"><span>edit</span></a><span 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principle</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:German_integral.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/German_integral.gif/50px-German_integral.gif" decoding="async" width="50" height="92" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/German_integral.gif/75px-German_integral.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/b/b5/German_integral.gif 2x" data-file-width="84" data-file-height="155" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related branches</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/Nonstandard_analysis" title="Nonstandard analysis">Nonstandard analysis</a></li> <li><a href="/wiki/Nonstandard_calculus" title="Nonstandard calculus">Nonstandard calculus</a></li> <li><a href="/wiki/Internal_set_theory" title="Internal set theory">Internal set theory</a></li> <li><a href="/wiki/Synthetic_differential_geometry" title="Synthetic differential geometry">Synthetic differential geometry</a></li> <li><a href="/wiki/Smooth_infinitesimal_analysis" title="Smooth infinitesimal analysis">Smooth infinitesimal analysis</a></li> <li><a href="/wiki/Constructive_nonstandard_analysis" title="Constructive nonstandard analysis">Constructive nonstandard analysis</a></li> <li><a href="/wiki/Infinitesimal_strain_theory" title="Infinitesimal strain theory">Infinitesimal strain theory (physics)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Formalizations</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/Differential_(mathematics)" title="Differential (mathematics)">Differentials</a></li> <li><a href="/wiki/Hyperreal_number" title="Hyperreal number">Hyperreal numbers</a></li> <li><a href="/wiki/Dual_number" title="Dual number">Dual numbers</a></li> <li><a href="/wiki/Surreal_number" title="Surreal number">Surreal numbers</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Individual concepts</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/Standard_part_function" title="Standard part function">Standard part function</a></li> <li><a href="/wiki/Transfer_principle" title="Transfer principle">Transfer principle</a></li> <li><a href="/wiki/Hyperinteger" title="Hyperinteger">Hyperinteger</a></li> <li><a href="/wiki/Increment_theorem" title="Increment theorem">Increment theorem</a></li> <li><a href="/wiki/Monad_(nonstandard_analysis)" title="Monad (nonstandard analysis)">Monad</a></li> <li><a href="/wiki/Internal_set" title="Internal set">Internal set</a></li> <li><a href="/wiki/Levi-Civita_field" title="Levi-Civita field">Levi-Civita field</a></li> <li><a href="/wiki/Hyperfinite_set" title="Hyperfinite set">Hyperfinite set</a></li> <li><a href="/wiki/Law_of_continuity" title="Law of continuity">Law of continuity</a></li> <li><a href="/wiki/Overspill" title="Overspill">Overspill</a></li> <li><a href="/wiki/Microcontinuity" title="Microcontinuity">Microcontinuity</a></li> <li><a href="/wiki/Transcendental_law_of_homogeneity" title="Transcendental law of homogeneity">Transcendental law of homogeneity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mathematicians</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/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a></li> <li><a href="/wiki/Abraham_Robinson" title="Abraham Robinson">Abraham Robinson</a></li> <li><a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Textbooks</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0;font-style:italic;"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Analyse_des_Infiniment_Petits_pour_l%27Intelligence_des_Lignes_Courbes" title="Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes">Analyse des Infiniment Petits</a></li> <li><a href="/wiki/Elementary_Calculus:_An_Infinitesimal_Approach" title="Elementary Calculus: An Infinitesimal Approach">Elementary Calculus</a></li> <li><a href="/wiki/Cours_d%27Analyse" 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