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और रंग बदलें" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="दिखावट" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-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-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">दिखावट</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_hi.wikipedia.org&uselang=hi" class=""><span>दान करें</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%96%E0%A4%BE%E0%A4%A4%E0%A4%BE_%E0%A4%AC%E0%A4%A8%E0%A4%BE%E0%A4%8F%E0%A4%81&returnto=%E0%A4%B0%E0%A5%80%E0%A4%AE%E0%A4%BE%E0%A4%A8-%E0%A4%B8%E0%A4%AE%E0%A4%BE%E0%A4%95%E0%A4%B2" title="हमारा सुझाव है की आप खाता बनाएँ और लॉगिन करें, परन्तु यह अनिवार्य नहीं है" class=""><span>खाता बनाएँ</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%B2%E0%A5%89%E0%A4%97%E0%A4%BF%E0%A4%A8&returnto=%E0%A4%B0%E0%A5%80%E0%A4%AE%E0%A4%BE%E0%A4%A8-%E0%A4%B8%E0%A4%AE%E0%A4%BE%E0%A4%95%E0%A4%B2" title="आपको सत्रारम्भ करने के लिए प्रोत्साहित किया जाता है; लेकिन यह अनिवार्य नहीं है [o]" accesskey="o" class=""><span>लॉग-इन करें</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="अन्य विकल्प" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="व्यक्तिगत उपकरण" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">व्यक्तिगत उपकरण</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_hi.wikipedia.org&uselang=hi"><span>दान करें</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%96%E0%A4%BE%E0%A4%A4%E0%A4%BE_%E0%A4%AC%E0%A4%A8%E0%A4%BE%E0%A4%8F%E0%A4%81&returnto=%E0%A4%B0%E0%A5%80%E0%A4%AE%E0%A4%BE%E0%A4%A8-%E0%A4%B8%E0%A4%AE%E0%A4%BE%E0%A4%95%E0%A4%B2" title="हमारा सुझाव है की आप खाता बनाएँ और लॉगिन करें, परन्तु यह अनिवार्य नहीं है"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>खाता बनाएँ</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%B2%E0%A5%89%E0%A4%97%E0%A4%BF%E0%A4%A8&returnto=%E0%A4%B0%E0%A5%80%E0%A4%AE%E0%A4%BE%E0%A4%A8-%E0%A4%B8%E0%A4%AE%E0%A4%BE%E0%A4%95%E0%A4%B2" title="आपको सत्रारम्भ करने के लिए प्रोत्साहित किया जाता है; लेकिन यह अनिवार्य नहीं है [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>लॉग-इन करें</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> लॉग-आउट किए गए संपादकों के लिए पृष्ठ <a href="/wiki/%E0%A4%B8%E0%A4%B9%E0%A4%BE%E0%A4%AF%E0%A4%A4%E0%A4%BE:%E0%A4%AA%E0%A4%B0%E0%A4%BF%E0%A4%9A%E0%A4%AF" aria-label="संपादन के बारे में अधिक जानें"><span>अधिक जानें</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%AE%E0%A5%87%E0%A4%B0%E0%A5%87_%E0%A4%AF%E0%A5%8B%E0%A4%97%E0%A4%A6%E0%A4%BE%E0%A4%A8" title="इस आईपी पते से संपादन की सूची [y]" accesskey="y"><span>योगदान</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%AE%E0%A5%87%E0%A4%B0%E0%A5%80_%E0%A4%B5%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%A4%E0%A4%BE" title="इस आईपी एड्रेससे हुए बदलावों के बारे में वार्ता [n]" accesskey="n"><span>वार्ता</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="साइट"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="सामग्री" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">सामग्री</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">साइडबार पर जाएँ</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">छुपाएँ</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">शुरुआत</div> </a> </li> <li id="toc-अवलोकन" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#अवलोकन"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>अवलोकन</span> </div> </a> <ul id="toc-अवलोकन-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-सन्दर्भ" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#सन्दर्भ"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>सन्दर्भ</span> </div> </a> <ul id="toc-सन्दर्भ-sublist" class="vector-toc-list"> </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="सामग्री" 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="विषयसूची को टॉगल करें" > <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">विषयसूची को टॉगल करें</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">रीमान-समाकल</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="दूसरे भाषा में लेख पर जाएँ। 38 भाषाओं में उपलब्ध" > <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-38" 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">38 भाषाएँ</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <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_%D8%B1%D9%8A%D9%85%D8%A7%D9%86" title="تكامل ريمان – अरबी" lang="ar" hreflang="ar" data-title="تكامل ريمان" data-language-autonym="العربية" data-language-local-name="अरबी" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Riman_inteqral%C4%B1" title="Riman inteqralı – अज़रबैजानी" lang="az" hreflang="az" data-title="Riman inteqralı" data-language-autonym="Azərbaycanca" data-language-local-name="अज़रबैजानी" class="interlanguage-link-target"><span>Azərbaycanca</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_%D0%A0%D1%8B%D0%BC%D0%B0%D0%BD%D0%B0" title="Інтэграл Рымана – बेलारूसी" lang="be" hreflang="be" data-title="Інтэграл Рымана" data-language-autonym="Беларуская" data-language-local-name="बेलारूसी" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Integral_de_Riemann" title="Integral de Riemann – कातालान" lang="ca" hreflang="ca" data-title="Integral de Riemann" data-language-autonym="Català" data-language-local-name="कातालान" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%AA%DB%95%D9%88%D8%A7%D9%88%DA%A9%D8%A7%D8%B1%DB%8C%DB%8C_%DA%95%DB%8C%D9%85%D8%A7%D9%86" title="تەواوکاریی ڕیمان – सोरानी कुर्दिश" lang="ckb" hreflang="ckb" data-title="تەواوکاریی ڕیمان" data-language-autonym="کوردی" data-language-local-name="सोरानी कुर्दिश" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Riemann%C5%AFv_integr%C3%A1l" title="Riemannův integrál – चेक" lang="cs" hreflang="cs" data-title="Riemannův integrál" data-language-autonym="Čeština" data-language-local-name="चेक" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A0%D0%B8%D0%BC%D0%B0%D0%BD_%D0%B8%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB%C4%95" title="Риман интегралĕ – चूवाश" lang="cv" hreflang="cv" data-title="Риман интегралĕ" data-language-autonym="Чӑвашла" data-language-local-name="चूवाश" 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/Riemannsches_Integral" title="Riemannsches Integral – जर्मन" lang="de" hreflang="de" data-title="Riemannsches Integral" data-language-autonym="Deutsch" data-language-local-name="जर्मन" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Riemann_integral" title="Riemann integral – अंग्रेज़ी" lang="en" hreflang="en" data-title="Riemann integral" data-language-autonym="English" data-language-local-name="अंग्रेज़ी" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Rimana_integralo" title="Rimana integralo – एस्पेरेंतो" lang="eo" hreflang="eo" data-title="Rimana integralo" data-language-autonym="Esperanto" data-language-local-name="एस्पेरेंतो" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Integral_de_Riemann" title="Integral de Riemann – स्पेनिश" lang="es" hreflang="es" data-title="Integral de Riemann" data-language-autonym="Español" data-language-local-name="स्पेनिश" 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/Riemannen_integral" title="Riemannen integral – बास्क" lang="eu" hreflang="eu" data-title="Riemannen integral" data-language-autonym="Euskara" data-language-local-name="बास्क" 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_%D8%B1%DB%8C%D9%85%D8%A7%D9%86" title="انتگرال ریمان – फ़ारसी" lang="fa" hreflang="fa" data-title="انتگرال ریمان" data-language-autonym="فارسی" data-language-local-name="फ़ारसी" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Riemannin_integraali" title="Riemannin integraali – फ़िनिश" lang="fi" hreflang="fi" data-title="Riemannin integraali" data-language-autonym="Suomi" data-language-local-name="फ़िनिश" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Int%C3%A9grale_de_Riemann" title="Intégrale de Riemann – फ़्रेंच" lang="fr" hreflang="fr" data-title="Intégrale de Riemann" data-language-autonym="Français" data-language-local-name="फ़्रेंच" 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_de_Riemann" title="Integral de Riemann – गैलिशियन" lang="gl" hreflang="gl" data-title="Integral de Riemann" data-language-autonym="Galego" data-language-local-name="गैलिशियन" class="interlanguage-link-target"><span>Galego</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_%D7%A8%D7%99%D7%9E%D7%9F" title="אינטגרל רימן – हिब्रू" lang="he" hreflang="he" data-title="אינטגרל רימן" data-language-autonym="עברית" data-language-local-name="हिब्रू" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Riemann-integr%C3%A1l" title="Riemann-integrál – हंगेरियाई" lang="hu" hreflang="hu" data-title="Riemann-integrál" data-language-autonym="Magyar" data-language-local-name="हंगेरियाई" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8C%D5%AB%D5%B4%D5%A1%D5%B6%D5%AB_%D5%AB%D5%B6%D5%BF%D5%A5%D5%A3%D6%80%D5%A1%D5%AC" title="Ռիմանի ինտեգրալ – आर्मेनियाई" lang="hy" hreflang="hy" data-title="Ռիմանի ինտեգրալ" data-language-autonym="Հայերեն" data-language-local-name="आर्मेनियाई" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Integral_Riemann" title="Integral Riemann – इंडोनेशियाई" lang="id" hreflang="id" data-title="Integral Riemann" data-language-autonym="Bahasa Indonesia" data-language-local-name="इंडोनेशियाई" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Integrale_di_Riemann" title="Integrale di Riemann – इतालवी" lang="it" hreflang="it" data-title="Integrale di Riemann" data-language-autonym="Italiano" data-language-local-name="इतालवी" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%AA%E3%83%BC%E3%83%9E%E3%83%B3%E7%A9%8D%E5%88%86" title="リーマン積分 – जापानी" lang="ja" hreflang="ja" data-title="リーマン積分" data-language-autonym="日本語" data-language-local-name="जापानी" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%A6%AC%EB%A7%8C_%EC%A0%81%EB%B6%84" title="리만 적분 – कोरियाई" lang="ko" hreflang="ko" data-title="리만 적분" data-language-autonym="한국어" data-language-local-name="कोरियाई" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Rymano_integralas" title="Rymano integralas – लिथुआनियाई" lang="lt" hreflang="lt" data-title="Rymano integralas" data-language-autonym="Lietuvių" data-language-local-name="लिथुआनियाई" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/R%C4%ABma%C5%86a_integr%C4%81lis" title="Rīmaņa integrālis – लातवियाई" lang="lv" hreflang="lv" data-title="Rīmaņa integrālis" data-language-autonym="Latviešu" data-language-local-name="लातवियाई" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Riemannintegratie" title="Riemannintegratie – डच" lang="nl" hreflang="nl" data-title="Riemannintegratie" data-language-autonym="Nederlands" data-language-local-name="डच" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Ca%C5%82ka_Riemanna" title="Całka Riemanna – पोलिश" lang="pl" hreflang="pl" data-title="Całka Riemanna" data-language-autonym="Polski" data-language-local-name="पोलिश" 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_de_Riemann" title="Integral de Riemann – पुर्तगाली" lang="pt" hreflang="pt" data-title="Integral de Riemann" data-language-autonym="Português" data-language-local-name="पुर्तगाली" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Integral%C4%83_Riemann" title="Integrală Riemann – रोमानियाई" lang="ro" hreflang="ro" data-title="Integrală Riemann" data-language-autonym="Română" data-language-local-name="रोमानियाई" 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_%D0%A0%D0%B8%D0%BC%D0%B0%D0%BD%D0%B0" title="Интеграл Римана – रूसी" lang="ru" hreflang="ru" data-title="Интеграл Римана" data-language-autonym="Русский" data-language-local-name="रूसी" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Ntigrali_di_Riemann" title="Ntigrali di Riemann – सिसिलियन" lang="scn" hreflang="scn" data-title="Ntigrali di Riemann" data-language-autonym="Sicilianu" data-language-local-name="सिसिलियन" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Riemannov_integr%C3%A1l" title="Riemannov integrál – स्लोवाक" lang="sk" hreflang="sk" 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href="https://tr.wikipedia.org/wiki/Riemann_integrali" title="Riemann integrali – तुर्की" lang="tr" hreflang="tr" data-title="Riemann integrali" data-language-autonym="Türkçe" data-language-local-name="तुर्की" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%86%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB_%D0%A0%D1%96%D0%BC%D0%B0%D0%BD%D0%B0" title="Інтеграл Рімана – यूक्रेनियाई" lang="uk" hreflang="uk" data-title="Інтеграл Рімана" data-language-autonym="Українська" data-language-local-name="यूक्रेनियाई" 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/%E9%BB%8E%E6%9B%BC%E7%A7%AF%E5%88%86" title="黎曼积分 – चीनी" lang="zh" hreflang="zh" data-title="黎曼积分" data-language-autonym="中文" data-language-local-name="चीनी" 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class="mw-indicators"> </div> <div id="siteSub" class="noprint">मुक्त ज्ञानकोश विकिपीडिया से</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="hi" dir="ltr"><figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/%E0%A4%9A%E0%A4%BF%E0%A4%A4%E0%A5%8D%E0%A4%B0:Integral_as_region_under_curve.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Integral_as_region_under_curve.svg/220px-Integral_as_region_under_curve.svg.png" decoding="async" width="220" height="205" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Integral_as_region_under_curve.svg/330px-Integral_as_region_under_curve.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Integral_as_region_under_curve.svg/440px-Integral_as_region_under_curve.svg.png 2x" data-file-width="750" data-file-height="700" /></a><figcaption>वक्र के नीचे स्थित क्षेत्र के क्षेत्रफल के रूप में समाकलन</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/%E0%A4%9A%E0%A4%BF%E0%A4%A4%E0%A5%8D%E0%A4%B0:Riemann_integral_regular.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Riemann_integral_regular.gif/220px-Riemann_integral_regular.gif" decoding="async" width="220" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Riemann_integral_regular.gif/330px-Riemann_integral_regular.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/28/Riemann_integral_regular.gif/440px-Riemann_integral_regular.gif 2x" data-file-width="558" data-file-height="300" /></a><figcaption>किसी अन्तराल के नियमित विभाजन के रीमान संकलनों का अनुक्रम। उपर प्रदर्शित अंक आयतों के कुल क्षेत्रफल को प्रदर्शित कर रहा है जो फलन के समाकल पर अभिसरित होता है।</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/%E0%A4%9A%E0%A4%BF%E0%A4%A4%E0%A5%8D%E0%A4%B0:Riemann_integral_irregular.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Riemann_integral_irregular.gif/220px-Riemann_integral_irregular.gif" decoding="async" width="220" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Riemann_integral_irregular.gif/330px-Riemann_integral_irregular.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Riemann_integral_irregular.gif/440px-Riemann_integral_irregular.gif 2x" data-file-width="558" data-file-height="300" /></a><figcaption>इस चित्र में प्रदर्शित विधि के अनुसार विभाजन का नियमित होना आवश्यक नहीं है। जब प्रत्येक उपखण्ड शून्य की ओर अग्रसर होता है तो सन्निकटन कार्य करता है।</figcaption></figure> <table class="vertical-navbox nowraplinks" style="float:right;clear:right;width:22.0em;margin:0 0 1.0em 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करें">वा</abbr></a></li><li class="nv-edit"><a class="external text" href="https://hi.wikipedia.org/w/index.php?title=%E0%A4%B8%E0%A4%BE%E0%A4%81%E0%A4%9A%E0%A4%BE:%E0%A4%95%E0%A4%B2%E0%A4%A8&action=edit"><abbr title="यह साँचा संपादित करें">सं</abbr></a></li></ul></div></td></tr></tbody></table> <p><a href="/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="गणित">गणित</a> की <a href="/wiki/%E0%A4%B5%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A4%B5%E0%A4%BF%E0%A4%95_%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%8D%E0%A4%B2%E0%A5%87%E0%A4%B7%E0%A4%A3" title="वास्तविक विश्लेषण">वास्तविक विश्लेषण</a> के रूप में पहचानी जाने वाली शाखा में <b>रीमान समाकलन</b> किसी <a href="/wiki/%E0%A4%AB%E0%A4%B2%E0%A4%A8" title="फलन">फलन</a> का किसी अन्तराल में परिभाषित प्रथम निश्चित परिभाषा है। यह परिभाषा <a href="/wiki/%E0%A4%AC%E0%A4%B0%E0%A5%8D%E0%A4%A8%E0%A4%B9%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%A1_%E0%A4%B0%E0%A5%80%E0%A4%AE%E0%A4%BE%E0%A4%A8" title="बर्नहार्ड रीमान">बर्नहार्ड रीमान</a> ने दी थी।<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> विभिन्न फलनोम और प्रायोगिक अनुप्रयोगों के लिए रीमान समाकलन <a href="/wiki/%E0%A4%95%E0%A4%B2%E0%A4%A8_%E0%A4%95%E0%A4%BE_%E0%A4%AE%E0%A5%82%E0%A4%B2%E0%A4%AD%E0%A5%82%E0%A4%A4_%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A5%87%E0%A4%AF" title="कलन का मूलभूत प्रमेय">कलन की मूलभूत प्रमेय</a> अथवा <a href="/wiki/%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%A4%E0%A5%8D%E0%A4%AE%E0%A4%95_%E0%A4%B8%E0%A4%AE%E0%A4%BE%E0%A4%95%E0%A4%B2%E0%A4%A8" title="संख्यात्मक समाकलन">संख्यात्मक समाकलन</a> के सन्निकटन द्वारा ज्ञात किया जा सकता है। </p><p>रीमान समाकलन विभिन्न सैद्धान्तिक उद्देश्यों के लिए अनुपयुक्त है। </p> <div class="mw-heading mw-heading2"><h2 id="अवलोकन"><span id=".E0.A4.85.E0.A4.B5.E0.A4.B2.E0.A5.8B.E0.A4.95.E0.A4.A8"></span>अवलोकन</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%A4%B0%E0%A5%80%E0%A4%AE%E0%A4%BE%E0%A4%A8-%E0%A4%B8%E0%A4%AE%E0%A4%BE%E0%A4%95%E0%A4%B2&action=edit&section=1" title="अनुभाग सम्पादित करें: अवलोकन"><span>संपादित करें</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>माना <i>f</i> अन्तराल [<i>a</i>, <i>b</i>] में धनात्मक <a href="/wiki/%E0%A4%B5%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A4%B5%E0%A4%BF%E0%A4%95_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="वास्तविक संख्या">वास्तविक-मान</a> फलन है और माना </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 S=\left\{(x,y)\,:\ a\leq x\leq b,0<y<f(x)\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>:</mo> <mtext> </mtext> <mi>a</mi> <mo>≤</mo> <mi>x</mi> <mo>≤</mo> <mi>b</mi> <mo>,</mo> <mn>0</mn> <mo><</mo> <mi>y</mi> <mo><</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\left\{(x,y)\,:\ a\leq x\leq b,0<y<f(x)\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73703316cbb6a6831715f9c85d63322446adf9c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.876ex; height:2.843ex;" alt="{\displaystyle S=\left\{(x,y)\,:\ a\leq x\leq b,0<y<f(x)\right\}}"></span></dd></dl> <p>अन्तराल [<i>a</i>, <i>b</i>] के उपर एवं फलन <i>f</i> से बनने वाले आरेख के नीचे का समतल क्षेत्र है (उपर दायें स्थित चित्र देखें)। हम <i>S</i> का क्षेत्रफल ज्ञात करना चाहते हैं। हमने एक बार इसका मापन किया और क्षेत्रफल को निम्नलिखित रूप में निरुपित करें: </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>रीमान समाकलन का मूल विचार <i>S</i> का क्षेत्रफल ज्ञात करने के लिए सरलतम सन्निकटनों का उपयोग करना है। अच्छे से अच्छा सन्निकटन लेने से हमारा अर्थ है कि <i>सीमान्त मान में</i> वक्र के अन्तर्गत <i>S</i> के क्षेत्रफल के पूरी तरह से समान है। </p><p>ध्यान रहे कि यहाँ <i>f</i> धनात्मक अथवा ऋणात्मक हो सकता है, <i>S</i> की परिभाषा को संशोधित किया जाता है जिससे समाकलन का मान आरेख के अन्दर आने वाले चिह्नित क्षेत्रफल के अनुरूप हो अर्थात <i>x</i>-अक्ष के उपर का क्षेत्रफल x-अक्ष के नीचे नीचे के क्षेत्रफल में से घटाने के तुल्य हो। </p> <div class="mw-heading mw-heading2"><h2 id="सन्दर्भ"><span id=".E0.A4.B8.E0.A4.A8.E0.A5.8D.E0.A4.A6.E0.A4.B0.E0.A5.8D.E0.A4.AD"></span>सन्दर्भ</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%A4%B0%E0%A5%80%E0%A4%AE%E0%A4%BE%E0%A4%A8-%E0%A4%B8%E0%A4%AE%E0%A4%BE%E0%A4%95%E0%A4%B2&action=edit&section=2" title="अनुभाग सम्पादित करें: सन्दर्भ"><span>संपादित करें</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="list-style-type: decimal;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">रीमान समाकलन की <a href="/wiki/%E0%A4%AC%E0%A4%B0%E0%A5%8D%E0%A4%A8%E0%A4%B9%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%A1_%E0%A4%B0%E0%A5%80%E0%A4%AE%E0%A4%BE%E0%A4%A8" title="बर्नहार्ड रीमान">बर्नहार्ड रीमान</a> ने अपने शोध पत्र "Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe" (त्रिकोणमितीय श्रेणियों के फलन के रूप में निरुपण; उदाहरण के लिए, किसी फलन को त्रिकोणमितीय श्रेणी के रूप में कब प्रदर्शित किया जा सकता है) में प्रस्तावित किया गया ह्ता। यह शोध पत्र गोट्टिंगन विश्वविद्यालय में 1854 में रीमान ने <i>Habilitationsschrift</i> (प्रशिक्षक बनने के लिए योग्यता प्राप्त करने हेतु) के रूप में प्रस्तुत किया था। यह 1868 में <i>प्रोसीडिंग्स ऑफ़ द रॉयल फिलोसोफिकल सोसाइटी, गौटिंगेन</i> (<i>Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen</i>) में प्रकाशित हुआ, संस्करण 13, पृष्ठ 87-132, (<a rel="nofollow" class="external text" href="http://books.google.com/books?id=PDVFAAAAcAAJ&pg=RA1-PA87">यहाँ</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150321202531/http://books.google.com/books?id=PDVFAAAAcAAJ&pg=RA1-PA87">Archived</a> 2015-03-21 at the <a href="/wiki/%E0%A4%B5%E0%A5%87%E0%A4%AC%E0%A5%88%E0%A4%95_%E0%A4%AE%E0%A4%B6%E0%A5%80%E0%A4%A8" title="वेबैक मशीन">वेबैक मशीन</a> पर ऑनलाइन उपलब्ध)। समाकलन के लिए रीमान की परिभाषा के लिए अनुभाग 4 देखें, "Über der Begriff eines bestimmten Integrals und den Umfang seiner Gültigkeit" (निश्चित समाकलन की अवधारणा और इसकी वैधता की सीमा), पृष्ठ 101-103.</span> </li> </ol></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: 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