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[r]" accesskey="r"><span>Perubahan terbaru</span></a></li><li id="n-Artikel-pilihan" class="mw-list-item"><a href="/wiki/Wikipedia:Artikel_pilihan/Topik"><span>Artikel pilihan</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Peristiwa_terkini" title="Temukan informasi tentang peristiwa terkini"><span>Peristiwa terkini</span></a></li><li id="n-newpage" class="mw-list-item"><a href="/wiki/Istimewa:Halaman_baru"><span>Halaman baru</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Istimewa:Halaman_sembarang" title="Tampilkan sembarang halaman [x]" accesskey="x"><span>Halaman sembarang</span></a></li><li id="n-specialpages" class="mw-list-item"><a href="/wiki/Istimewa:Halaman_istimewa"><span>Halaman istimewa</span></a></li> </ul> </div> </div> <div id="p-Komunitas" class="vector-menu mw-portlet mw-portlet-Komunitas" > <div class="vector-menu-heading"> Komunitas </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-Warung-Kopi" class="mw-list-item"><a href="/wiki/Wikipedia:Warung_Kopi"><span>Warung Kopi</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Portal:Komunitas" title="Tentang proyek, apa yang dapat Anda lakukan, di mana untuk mencari sesuatu"><span>Portal komunitas</span></a></li><li id="n-help" class="mw-list-item"><a href="/wiki/Bantuan:Isi" title="Tempat mencari bantuan."><span>Bantuan</span></a></li> </ul> </div> </div> <div id="p-Wikipedia" class="vector-menu mw-portlet mw-portlet-Wikipedia" > <div class="vector-menu-heading"> Wikipedia </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:Perihal"><span>Tentang Wikipedia</span></a></li><li id="n-Pancapilar" class="mw-list-item"><a href="/wiki/Wikipedia:Pancapilar"><span>Pancapilar</span></a></li><li id="n-Kebijakan" class="mw-list-item"><a href="/wiki/Wikipedia:Kebijakan_dan_pedoman"><span>Kebijakan</span></a></li><li id="n-Hubungi-kami" class="mw-list-item"><a href="/wiki/Wikipedia:Hubungi_kami"><span>Hubungi kami</span></a></li><li id="n-Bak-pasir" class="mw-list-item"><a href="/wiki/Wikipedia:Bak_pasir"><span>Bak pasir</span></a></li> </ul> </div> </div> <div id="p-Bagikan" class="vector-menu mw-portlet mw-portlet-Bagikan emptyPortlet" > <div class="vector-menu-heading"> Bagikan </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Halaman_Utama" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="Ensiklopedia Bebas" src="/static/images/mobile/copyright/wikipedia-tagline-id.svg" width="120" height="14" style="width: 7.5em; height: 0.875em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Istimewa:Pencarian" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Cari di Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Pencarian</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Telusuri Wikipedia" aria-label="Telusuri Wikipedia" autocapitalize="sentences" title="Cari di Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Istimewa:Pencarian"> </div> <button class="cdx-button cdx-search-input__end-button">Cari</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Perkakas pribadi"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" 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-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Tampilan"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <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="Tampilan" > <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">Tampilan</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="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=id.wikipedia.org&uselang=id" class=""><span>Menyumbang</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=Istimewa:Buat_akun&returnto=Kalkulus" title="Anda dianjurkan untuk membuat akun dan masuk log; meskipun, hal itu tidak diwajibkan" class=""><span>Buat akun baru</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=Istimewa:Masuk_log&returnto=Kalkulus" title="Anda disarankan untuk masuk log, meskipun hal itu tidak diwajibkan. [o]" accesskey="o" class=""><span>Masuk log</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="Opsi lainnya" > <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="Perkakas pribadi" > <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">Perkakas pribadi</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Menu pengguna" > <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="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=id.wikipedia.org&uselang=id"><span>Menyumbang</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Istimewa:Buat_akun&returnto=Kalkulus" title="Anda dianjurkan untuk membuat akun dan masuk log; meskipun, hal itu tidak diwajibkan"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Buat akun baru</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Istimewa:Masuk_log&returnto=Kalkulus" title="Anda disarankan untuk masuk log, meskipun hal itu tidak diwajibkan. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Masuk log</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"> Halaman penyunting yang telah keluar log <a href="/wiki/Bantuan:Pengantar" aria-label="Pelajari lebih lanjut tentang menyunting"><span>pelajari lebih lanjut</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/Istimewa:Kontribusi_saya" title="Daftar suntingan yang dibuat dari alamat IP ini [y]" accesskey="y"><span>Kontribusi</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Istimewa:Pembicaraan_saya" title="Pembicaraan tentang suntingan dari alamat IP ini [n]" accesskey="n"><span>Pembicaraan</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="Situs"> <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="Daftar isi" 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">Daftar isi</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">pindah ke bilah sisi</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">sembunyikan</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">Awal</div> </a> </li> <li id="toc-Sejarah" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sejarah"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Sejarah</span> </div> </a> <button aria-controls="toc-Sejarah-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>Gulingkan subbagian Sejarah</span> </button> <ul id="toc-Sejarah-sublist" class="vector-toc-list"> <li id="toc-Perkembangan" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Perkembangan"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Perkembangan</span> </div> </a> <ul id="toc-Perkembangan-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pengaruh_penting" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pengaruh_penting"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Pengaruh penting</span> </div> </a> <ul id="toc-Pengaruh_penting-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Prinsip_dasar" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Prinsip_dasar"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Prinsip dasar</span> </div> </a> <button aria-controls="toc-Prinsip_dasar-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>Gulingkan subbagian Prinsip dasar</span> </button> <ul id="toc-Prinsip_dasar-sublist" class="vector-toc-list"> <li id="toc-Limit_dan_kecil_tak_terhingga" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Limit_dan_kecil_tak_terhingga"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Limit dan kecil tak terhingga</span> </div> </a> <ul id="toc-Limit_dan_kecil_tak_terhingga-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Turunan" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Turunan"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Turunan</span> </div> </a> <ul id="toc-Turunan-sublist" class="vector-toc-list"> <li id="toc-Notasi_pendiferensialan" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Notasi_pendiferensialan"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Notasi pendiferensialan</span> </div> </a> <ul id="toc-Notasi_pendiferensialan-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Integral" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integral"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Integral</span> </div> </a> <ul id="toc-Integral-sublist" class="vector-toc-list"> <li id="toc-Integral_tertentu" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Integral_tertentu"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.1</span> <span>Integral tertentu</span> </div> </a> <ul id="toc-Integral_tertentu-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Integral_tak_tentu" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Integral_tak_tentu"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.2</span> <span>Integral tak tentu</span> </div> </a> <ul id="toc-Integral_tak_tentu-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Teorema_dasar" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Teorema_dasar"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Teorema dasar</span> </div> </a> <ul id="toc-Teorema_dasar-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Aplikasi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Aplikasi"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Aplikasi</span> </div> </a> <ul id="toc-Aplikasi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lihat_pula" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lihat_pula"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Lihat pula</span> </div> </a> <ul id="toc-Lihat_pula-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referensi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referensi"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Referensi</span> </div> </a> <ul id="toc-Referensi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Daftar_pustaka" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Daftar_pustaka"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Daftar pustaka</span> </div> </a> <ul id="toc-Daftar_pustaka-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sumber_lain" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sumber_lain"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Sumber lain</span> </div> </a> <button aria-controls="toc-Sumber_lain-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>Gulingkan subbagian Sumber lain</span> </button> <ul id="toc-Sumber_lain-sublist" class="vector-toc-list"> <li id="toc-Bacaan_lebih_lanjut" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bacaan_lebih_lanjut"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Bacaan lebih lanjut</span> </div> </a> <ul id="toc-Bacaan_lebih_lanjut-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pustaka_daring" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pustaka_daring"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Pustaka daring</span> </div> </a> <ul id="toc-Pustaka_daring-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Halaman_web" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Halaman_web"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>Halaman web</span> </div> </a> <ul id="toc-Halaman_web-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="Daftar isi" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Daftar Isi" > <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="Gulingkan daftar isi" > <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">Gulingkan daftar isi</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">Kalkulus</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="Pergi ke artikel dalam bahasa lain. Terdapat 97 bahasa" > <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-97" 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">97 bahasa</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%AB%E1%88%8D%E1%8A%A9%E1%88%88%E1%88%B5" title="ካልኩለስ – Amharik" lang="am" hreflang="am" data-title="ካልኩለስ" data-language-autonym="አማርኛ" data-language-local-name="Amharik" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Calculo" title="Calculo – Aragon" lang="an" hreflang="an" data-title="Calculo" data-language-autonym="Aragonés" data-language-local-name="Aragon" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D9%81%D8%A7%D8%B6%D9%84_%D9%88%D8%AA%D9%83%D8%A7%D9%85%D9%84" title="تفاضل وتكامل – Arab" lang="ar" hreflang="ar" data-title="تفاضل وتكامل" data-language-autonym="العربية" data-language-local-name="Arab" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D8%AA%D9%81%D8%A7%D8%B6%D9%84_%D9%88%D8%AA%D9%83%D8%A7%D9%85%D9%84" title="تفاضل وتكامل – Arab Mesir" lang="arz" hreflang="arz" data-title="تفاضل وتكامل" data-language-autonym="مصرى" data-language-local-name="Arab Mesir" class="interlanguage-link-target"><span>مصرى</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/C%C3%A1lculu" title="Cálculu – Asturia" lang="ast" hreflang="ast" data-title="Cálculu" data-language-autonym="Asturianu" data-language-local-name="Asturia" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Kalkulus" title="Kalkulus – Central Bikol" lang="bcl" hreflang="bcl" data-title="Kalkulus" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-bg badge-Q70894304 mw-list-item" title=""><a href="https://bg.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB%D0%BD%D0%BE_%D0%B8_%D0%B8%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB%D0%BD%D0%BE_%D1%81%D0%BC%D1%8F%D1%82%D0%B0%D0%BD%D0%B5" title="Диференциално и интегрално смятане – Bulgaria" lang="bg" hreflang="bg" data-title="Диференциално и интегрално смятане" data-language-autonym="Български" data-language-local-name="Bulgaria" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-blk mw-list-item"><a href="https://blk.wikipedia.org/wiki/%E1%80%80%E1%80%B2%E1%80%B8%E1%80%80%E1%80%AF%E1%80%9C%E1%80%90%E1%80%BA%E1%80%9E%E1%80%BA" title="ကဲးကုလတ်သ် – Pa'O" lang="blk" hreflang="blk" data-title="ကဲးကုလတ်သ်" data-language-autonym="ပအိုဝ်ႏဘာႏသာႏ" data-language-local-name="Pa'O" class="interlanguage-link-target"><span>ပအိုဝ်ႏဘာႏသာႏ</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%95%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%B2%E0%A6%95%E0%A7%81%E0%A6%B2%E0%A6%BE%E0%A6%B8" title="ক্যালকুলাস – Bengali" lang="bn" hreflang="bn" data-title="ক্যালকুলাস" data-language-autonym="বাংলা" data-language-local-name="Bengali" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Infinitezimalni_ra%C4%8Dun" title="Infinitezimalni račun – Bosnia" lang="bs" hreflang="bs" data-title="Infinitezimalni račun" data-language-autonym="Bosanski" data-language-local-name="Bosnia" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/C%C3%A0lcul_infinitesimal" title="Càlcul infinitesimal – Katalan" lang="ca" hreflang="ca" data-title="Càlcul infinitesimal" data-language-autonym="Català" data-language-local-name="Katalan" 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%AC%DB%8C%D8%A7%DA%A9%D8%A7%D8%B1%DB%8C_%D9%88_%D8%AA%DB%95%D9%88%D8%A7%D9%88%DA%A9%D8%A7%D8%B1%DB%8C" title="جیاکاری و تەواوکاری – Kurdi Sorani" lang="ckb" hreflang="ckb" data-title="جیاکاری و تەواوکاری" data-language-autonym="کوردی" data-language-local-name="Kurdi Sorani" 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/Infinitezim%C3%A1ln%C3%AD_po%C4%8Det" title="Infinitezimální počet – Ceko" lang="cs" hreflang="cs" data-title="Infinitezimální počet" data-language-autonym="Čeština" data-language-local-name="Ceko" 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%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%C4%83%D0%BB%D0%BB%D0%B0_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7" title="Математикăлла анализ – Chuvash" lang="cv" hreflang="cv" data-title="Математикăлла анализ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Calcwlws" title="Calcwlws – Welsh" lang="cy" hreflang="cy" data-title="Calcwlws" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Infinitesimalregning" title="Infinitesimalregning – Dansk" lang="da" hreflang="da" data-title="Infinitesimalregning" data-language-autonym="Dansk" data-language-local-name="Dansk" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de badge-Q70894304 mw-list-item" title=""><a href="https://de.wikipedia.org/wiki/Infinitesimalrechnung" title="Infinitesimalrechnung – Jerman" lang="de" hreflang="de" data-title="Infinitesimalrechnung" data-language-autonym="Deutsch" data-language-local-name="Jerman" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-diq mw-list-item"><a href="https://diq.wikipedia.org/wiki/Kalkulus" title="Kalkulus – Dimli" lang="diq" hreflang="diq" data-title="Kalkulus" data-language-autonym="Zazaki" data-language-local-name="Dimli" class="interlanguage-link-target"><span>Zazaki</span></a></li><li class="interlanguage-link interwiki-dtp mw-list-item"><a href="https://dtp.wikipedia.org/wiki/Sidsimban" title="Sidsimban – Central Dusun" lang="dtp" hreflang="dtp" data-title="Sidsimban" data-language-autonym="Kadazandusun" data-language-local-name="Central Dusun" class="interlanguage-link-target"><span>Kadazandusun</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9B%CE%BF%CE%B3%CE%B9%CF%83%CE%BC%CF%8C%CF%82" title="Λογισμός – Yunani" lang="el" hreflang="el" data-title="Λογισμός" data-language-autonym="Ελληνικά" data-language-local-name="Yunani" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Calculus" title="Calculus – Inggris" lang="en" hreflang="en" data-title="Calculus" data-language-autonym="English" data-language-local-name="Inggris" 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/Infinitezima_kalkulo" title="Infinitezima kalkulo – Esperanto" lang="eo" hreflang="eo" data-title="Infinitezima kalkulo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" 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/C%C3%A1lculo_infinitesimal" title="Cálculo infinitesimal – Spanyol" lang="es" hreflang="es" data-title="Cálculo infinitesimal" data-language-autonym="Español" data-language-local-name="Spanyol" 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/Kalkulu_infinitesimal" title="Kalkulu infinitesimal – Basque" lang="eu" hreflang="eu" data-title="Kalkulu infinitesimal" 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%AD%D8%B3%D8%A7%D8%A8%D8%A7%D9%86" title="حسابان – Persia" lang="fa" hreflang="fa" data-title="حسابان" data-language-autonym="فارسی" data-language-local-name="Persia" 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/Differentiaali-_ja_integraalilaskenta" title="Differentiaali- ja integraalilaskenta – Suomi" lang="fi" hreflang="fi" data-title="Differentiaali- ja integraalilaskenta" data-language-autonym="Suomi" data-language-local-name="Suomi" 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/Calcul_infinit%C3%A9simal" title="Calcul infinitésimal – Prancis" lang="fr" hreflang="fr" data-title="Calcul infinitésimal" data-language-autonym="Français" data-language-local-name="Prancis" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Calcalas" title="Calcalas – Irlandia" lang="ga" hreflang="ga" data-title="Calcalas" data-language-autonym="Gaeilge" data-language-local-name="Irlandia" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E5%BE%AE%E7%A9%8D%E5%88%86" title="微積分 – Gan" lang="gan" hreflang="gan" data-title="微積分" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/C%C3%A1lculo_infinitesimal" title="Cálculo infinitesimal – Galisia" lang="gl" hreflang="gl" data-title="Cálculo infinitesimal" data-language-autonym="Galego" data-language-local-name="Galisia" 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%95%E0%AA%B2%E0%AA%A8_%E0%AA%B6%E0%AA%BE%E0%AA%B8%E0%AB%8D%E0%AA%A4%E0%AB%8D%E0%AA%B0" title="કલન શાસ્ત્ર – Gujarat" lang="gu" hreflang="gu" data-title="કલન શાસ્ત્ર" data-language-autonym="ગુજરાતી" data-language-local-name="Gujarat" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-ha mw-list-item"><a href="https://ha.wikipedia.org/wiki/Calculus" title="Calculus – Hausa" lang="ha" hreflang="ha" data-title="Calculus" data-language-autonym="Hausa" data-language-local-name="Hausa" class="interlanguage-link-target"><span>Hausa</span></a></li><li class="interlanguage-link interwiki-hak mw-list-item"><a href="https://hak.wikipedia.org/wiki/M%C3%AC-chit-f%C3%BBn-ho%CC%8Dk" title="Mì-chit-fûn-ho̍k – Hakka Chinese" lang="hak" hreflang="hak" data-title="Mì-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-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%90%D7%99%D7%A0%D7%A4%D7%99%D7%A0%D7%99%D7%98%D7%A1%D7%99%D7%9E%D7%9C%D7%99" title="חשבון אינפיניטסימלי – Ibrani" lang="he" hreflang="he" data-title="חשבון אינפיניטסימלי" data-language-autonym="עברית" data-language-local-name="Ibrani" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%95%E0%A4%B2%E0%A4%A8" title="कलन – Hindi" lang="hi" hreflang="hi" data-title="कलन" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Calculus" title="Calculus – Hindi Fiji" lang="hif" hreflang="hif" data-title="Calculus" data-language-autonym="Fiji Hindi" data-language-local-name="Hindi Fiji" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Infinitezimalni_ra%C4%8Dun" title="Infinitezimalni račun – Kroasia" lang="hr" hreflang="hr" data-title="Infinitezimalni račun" data-language-autonym="Hrvatski" data-language-local-name="Kroasia" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Calculo_infinitesimal" title="Calculo infinitesimal – Interlingua" lang="ia" hreflang="ia" data-title="Calculo infinitesimal" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-iba mw-list-item"><a href="https://iba.wikipedia.org/wiki/Kalkulus" title="Kalkulus – Iban" lang="iba" hreflang="iba" data-title="Kalkulus" data-language-autonym="Jaku Iban" data-language-local-name="Iban" class="interlanguage-link-target"><span>Jaku Iban</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Kalkulo" title="Kalkulo – Ido" lang="io" hreflang="io" data-title="Kalkulo" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/%C3%96rsm%C3%A6%C3%B0areikningur" title="Örsmæðareikningur – Islandia" lang="is" hreflang="is" data-title="Örsmæðareikningur" data-language-autonym="Íslenska" data-language-local-name="Islandia" 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/Calcolo_infinitesimale" title="Calcolo infinitesimale – Italia" lang="it" hreflang="it" data-title="Calcolo infinitesimale" data-language-autonym="Italiano" data-language-local-name="Italia" 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/%E5%BE%AE%E5%88%86%E7%A9%8D%E5%88%86%E5%AD%A6" title="微分積分学 – Jepang" lang="ja" hreflang="ja" data-title="微分積分学" data-language-autonym="日本語" data-language-local-name="Jepang" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Kialkiulos" title="Kialkiulos – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Kialkiulos" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-jv badge-Q17437796 badge-featuredarticle mw-list-item" title="artikel pilihan"><a href="https://jv.wikipedia.org/wiki/Kalkulus" title="Kalkulus – Jawa" lang="jv" hreflang="jv" data-title="Kalkulus" data-language-autonym="Jawa" data-language-local-name="Jawa" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%99%E1%83%90%E1%83%9A%E1%83%99%E1%83%A3%E1%83%9A%E1%83%A3%E1%83%A1%E1%83%98" title="კალკულუსი – Georgia" lang="ka" hreflang="ka" data-title="კალკულუსი" data-language-autonym="ქართული" data-language-local-name="Georgia" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%AF%B8%EC%A0%81%EB%B6%84%ED%95%99" title="미적분학 – Korea" lang="ko" hreflang="ko" data-title="미적분학" data-language-autonym="한국어" data-language-local-name="Korea" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Calculus_infinitesimalis" title="Calculus infinitesimalis – Latin" lang="la" hreflang="la" data-title="Calculus infinitesimalis" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Calculo" title="Calculo – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Calculo" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Infinitesimaalraekening" title="Infinitesimaalraekening – Limburgia" lang="li" hreflang="li" data-title="Infinitesimaalraekening" data-language-autonym="Limburgs" data-language-local-name="Limburgia" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Integralinis_ir_diferencialinis_skai%C4%8Diavimas" title="Integralinis ir diferencialinis skaičiavimas – Lituania" lang="lt" hreflang="lt" data-title="Integralinis ir diferencialinis skaičiavimas" data-language-autonym="Lietuvių" data-language-local-name="Lituania" 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%93%C4%B7ini" title="Rēķini – Latvia" lang="lv" hreflang="lv" data-title="Rēķini" data-language-autonym="Latviešu" data-language-local-name="Latvia" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mad mw-list-item"><a href="https://mad.wikipedia.org/wiki/Kalkulus" title="Kalkulus – Madura" lang="mad" hreflang="mad" data-title="Kalkulus" data-language-autonym="Madhurâ" data-language-local-name="Madura" class="interlanguage-link-target"><span>Madhurâ</span></a></li><li class="interlanguage-link interwiki-min mw-list-item"><a href="https://min.wikipedia.org/wiki/Kalkulus" title="Kalkulus – Minangkabau" lang="min" hreflang="min" data-title="Kalkulus" data-language-autonym="Minangkabau" data-language-local-name="Minangkabau" class="interlanguage-link-target"><span>Minangkabau</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%98%D0%BD%D1%84%D0%B8%D0%BD%D0%B8%D1%82%D0%B5%D0%B7%D0%B8%D0%BC%D0%B0%D0%BB%D0%BD%D0%BE_%D1%81%D0%BC%D0%B5%D1%82%D0%B0%D1%9A%D0%B5" title="Инфинитезимално сметање – Makedonia" lang="mk" hreflang="mk" data-title="Инфинитезимално сметање" data-language-autonym="Македонски" data-language-local-name="Makedonia" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%95%E0%B4%B2%E0%B4%A8%E0%B4%82" title="കലനം – Malayalam" lang="ml" hreflang="ml" data-title="കലനം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</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_%D0%B1%D0%B0_%D0%B4%D0%B8%D1%84%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB_%D1%82%D0%BE%D0%BE%D0%BB%D0%BE%D0%BB" title="Интеграл ба дифференциал тоолол – Mongolia" lang="mn" hreflang="mn" data-title="Интеграл ба дифференциал тоолол" data-language-autonym="Монгол" data-language-local-name="Mongolia" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%95%E0%A4%B2%E0%A4%A8" title="कलन – Marathi" lang="mr" hreflang="mr" data-title="कलन" 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/Kalkulus" title="Kalkulus – Melayu" lang="ms" hreflang="ms" data-title="Kalkulus" data-language-autonym="Bahasa Melayu" data-language-local-name="Melayu" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%80%E1%80%B2%E1%80%80%E1%80%AF%E1%80%9C%E1%80%95%E1%80%BA" title="ကဲကုလပ် – Burma" lang="my" hreflang="my" data-title="ကဲကုလပ်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burma" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%95%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%B2%E0%A5%8D%E0%A4%95%E0%A5%81%E0%A4%B2%E0%A4%B8" title="क्याल्कुलस – Newari" lang="new" hreflang="new" data-title="क्याल्कुलस" data-language-autonym="नेपाल भाषा" data-language-local-name="Newari" class="interlanguage-link-target"><span>नेपाल भाषा</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Kalkulus" title="Kalkulus – Nynorsk Norwegia" lang="nn" hreflang="nn" data-title="Kalkulus" data-language-autonym="Norsk nynorsk" data-language-local-name="Nynorsk Norwegia" 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/Calcul_infinitesimal" title="Calcul infinitesimal – Ositania" lang="oc" hreflang="oc" data-title="Calcul infinitesimal" data-language-autonym="Occitan" data-language-local-name="Ositania" 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/Kaalkulasii" title="Kaalkulasii – Oromo" lang="om" hreflang="om" data-title="Kaalkulasii" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%95%E0%A9%88%E0%A8%B2%E0%A8%95%E0%A9%82%E0%A8%B2%E0%A8%B8" title="ਕੈਲਕੂਲਸ – Punjabi" lang="pa" hreflang="pa" data-title="ਕੈਲਕੂਲਸ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%DA%A9%DB%8C%D9%84%DA%A9%D9%88%D9%84%D8%B3" title="کیلکولس – Western Punjabi" lang="pnb" hreflang="pnb" data-title="کیلکولس" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/C%C3%A1lculo_infinitesimal" title="Cálculo infinitesimal – Portugis" lang="pt" hreflang="pt" data-title="Cálculo infinitesimal" data-language-autonym="Português" data-language-local-name="Portugis" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Yupaylliy" title="Yupaylliy – Quechua" lang="qu" hreflang="qu" data-title="Yupaylliy" data-language-autonym="Runa Simi" data-language-local-name="Quechua" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Calcul_infinitezimal" title="Calcul infinitezimal – Rumania" lang="ro" hreflang="ro" data-title="Calcul infinitezimal" data-language-autonym="Română" data-language-local-name="Rumania" 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%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B9_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7" title="Математический анализ – Rusia" lang="ru" hreflang="ru" data-title="Математический анализ" data-language-autonym="Русский" data-language-local-name="Rusia" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D0%B9_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7" title="Математическай анализ – Sakha" lang="sah" hreflang="sah" data-title="Математическай анализ" data-language-autonym="Саха тыла" data-language-local-name="Sakha" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Calculus" title="Calculus – Skotlandia" lang="sco" hreflang="sco" data-title="Calculus" data-language-autonym="Scots" data-language-local-name="Skotlandia" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Infinitezimalni_ra%C4%8Dun" title="Infinitezimalni račun – Serbo-Kroasia" lang="sh" hreflang="sh" data-title="Infinitezimalni račun" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Kroasia" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%9A%E0%B6%BD%E0%B6%B1%E0%B6%BA" title="කලනය – Sinhala" lang="si" hreflang="si" data-title="කලනය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Calculus" title="Calculus – Simple English" lang="en-simple" hreflang="en-simple" data-title="Calculus" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Diferenci%C3%A1lny_a_integr%C3%A1lny_po%C4%8Det" title="Diferenciálny a integrálny počet – Slovak" lang="sk" hreflang="sk" data-title="Diferenciálny a integrálny počet" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Infinitezimalni_ra%C4%8Dun" title="Infinitezimalni račun – Slovenia" lang="sl" hreflang="sl" data-title="Infinitezimalni račun" data-language-autonym="Slovenščina" data-language-local-name="Slovenia" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Infinitezimalni_ra%C4%8Dun" title="Infinitezimalni račun – Serbia" lang="sr" hreflang="sr" data-title="Infinitezimalni račun" data-language-autonym="Српски / srpski" data-language-local-name="Serbia" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-ss mw-list-item"><a href="https://ss.wikipedia.org/wiki/Calculus" title="Calculus – Swati" lang="ss" hreflang="ss" data-title="Calculus" data-language-autonym="SiSwati" data-language-local-name="Swati" class="interlanguage-link-target"><span>SiSwati</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Infinitesimalkalkyl" title="Infinitesimalkalkyl – Swedia" lang="sv" hreflang="sv" data-title="Infinitesimalkalkyl" data-language-autonym="Svenska" data-language-local-name="Swedia" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A8%E0%AF%81%E0%AE%A3%E0%AF%8D%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D" title="நுண்கணிதம் – Tamil" lang="ta" hreflang="ta" data-title="நுண்கணிதம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%95%E0%B0%B2%E0%B0%A8_%E0%B0%97%E0%B0%A3%E0%B0%BF%E0%B0%A4%E0%B0%AE%E0%B1%81" title="కలన గణితము – Telugu" lang="te" hreflang="te" data-title="కలన గణితము" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%81%E0%B8%84%E0%B8%A5%E0%B8%84%E0%B8%B9%E0%B8%A5%E0%B8%B1%E0%B8%AA" title="แคลคูลัส – Thai" lang="th" hreflang="th" data-title="แคลคูลัส" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Calculus" title="Calculus – Tagalog" lang="tl" hreflang="tl" data-title="Calculus" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Kalk%C3%BCl%C3%BCs" title="Kalkülüs – Turki" lang="tr" hreflang="tr" data-title="Kalkülüs" data-language-autonym="Türkçe" data-language-local-name="Turki" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tum mw-list-item"><a href="https://tum.wikipedia.org/wiki/Kakyula" title="Kakyula – Tumbuka" lang="tum" hreflang="tum" data-title="Kakyula" data-language-autonym="ChiTumbuka" data-language-local-name="Tumbuka" class="interlanguage-link-target"><span>ChiTumbuka</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B5_%D1%82%D0%B0_%D1%96%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%B5%D0%BD%D0%BD%D1%8F" title="Диференціальне та інтегральне числення – Ukraina" lang="uk" hreflang="uk" data-title="Диференціальне та інтегральне числення" data-language-autonym="Українська" data-language-local-name="Ukraina" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%A7%D8%AD%D8%B5%D8%A7" title="احصا – Urdu" lang="ur" hreflang="ur" data-title="احصا" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/C%C3%B3nto_infinitezima%C5%82e" title="Cónto infinitezimałe – Venesia" lang="vec" hreflang="vec" data-title="Cónto infinitezimałe" data-language-autonym="Vèneto" data-language-local-name="Venesia" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Vi_t%C3%ADch_ph%C3%A2n" title="Vi tích phân – Vietnam" lang="vi" hreflang="vi" data-title="Vi tích phân" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnam" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Kalkulo" title="Kalkulo – Warai" lang="war" hreflang="war" data-title="Kalkulo" data-language-autonym="Winaray" data-language-local-name="Warai" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%BE%AE%E7%A7%AF%E5%88%86%E5%AD%A6" title="微积分学 – Wu Tionghoa" lang="wuu" hreflang="wuu" data-title="微积分学" data-language-autonym="吴语" data-language-local-name="Wu Tionghoa" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A7%D7%90%D7%9C%D7%A7%D7%95%D7%9C%D7%95%D7%A1" title="קאלקולוס – Yiddish" lang="yi" hreflang="yi" data-title="קאלקולוס" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%BE%AE%E7%A7%AF%E5%88%86%E5%AD%A6" title="微积分学 – Tionghoa" lang="zh" hreflang="zh" data-title="微积分学" data-language-autonym="中文" data-language-local-name="Tionghoa" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%BE%AE%E7%A9%8D%E5%88%86" title="微積分 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="微積分" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/B%C3%AE-chek-hun" title="Bî-chek-hun – Minnan" lang="nan" hreflang="nan" data-title="Bî-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-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%BE%AE%E7%A9%8D%E5%88%86" title="微積分 – Kanton" lang="yue" hreflang="yue" data-title="微積分" data-language-autonym="粵語" data-language-local-name="Kanton" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q149972#sitelinks-wikipedia" title="Sunting pranala interwiki" class="wbc-editpage">Sunting pranala</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Ruang nama"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Kalkulus" title="Lihat halaman isi [c]" accesskey="c"><span>Halaman</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Pembicaraan:Kalkulus" rel="discussion" title="Pembicaraan halaman isi [t]" accesskey="t"><span>Pembicaraan</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Ubah varian bahasa" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Bahasa Indonesia</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Tampilan"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Kalkulus"><span>Baca</span></a></li><li id="ca-viewsource" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Kalkulus&action=edit" title="Halaman ini dilindungi. Anda hanya dapat melihat sumbernya. [e]" accesskey="e"><span>Lihat sumber</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Kalkulus&action=history" title="Revisi sebelumnya dari halaman ini. [h]" accesskey="h"><span>Lihat riwayat</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Peralatan halaman"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Perkakas" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Perkakas</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Perkakas</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">pindah ke bilah sisi</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">sembunyikan</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Opsi lainnya" > <div class="vector-menu-heading"> Tindakan </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Kalkulus"><span>Baca</span></a></li><li id="ca-more-viewsource" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Kalkulus&action=edit"><span>Lihat sumber</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Kalkulus&action=history"><span>Lihat riwayat</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Umum </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Istimewa:Pranala_balik/Kalkulus" title="Daftar semua halaman wiki yang memiliki pranala ke halaman ini [j]" accesskey="j"><span>Pranala balik</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Istimewa:Perubahan_terkait/Kalkulus" rel="nofollow" title="Perubahan terbaru halaman-halaman yang memiliki pranala ke halaman ini [k]" accesskey="k"><span>Perubahan terkait</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Kalkulus&oldid=26879836" title="Pranala permanen untuk revisi halaman ini"><span>Pranala permanen</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Kalkulus&action=info" title="Informasi lanjut tentang halaman ini"><span>Informasi halaman</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Istimewa:Kutip&page=Kalkulus&id=26879836&wpFormIdentifier=titleform" title="Informasi tentang bagaimana mengutip halaman ini"><span>Kutip halaman ini</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Istimewa:UrlShortener&url=https%3A%2F%2Fid.wikipedia.org%2Fwiki%2FKalkulus"><span>Lihat URL pendek</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Istimewa:QrCode&url=https%3A%2F%2Fid.wikipedia.org%2Fwiki%2FKalkulus"><span>Unduh kode QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Cetak/ekspor </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Istimewa:Buku&bookcmd=book_creator&referer=Kalkulus"><span>Buat buku</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Istimewa:DownloadAsPdf&page=Kalkulus&action=show-download-screen"><span>Unduh versi PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Kalkulus&printable=yes" title="Versi cetak halaman ini [p]" accesskey="p"><span>Versi cetak</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> Dalam proyek lain </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Calculus" hreflang="en"><span>Wikimedia Commons</span></a></li><li class="wb-otherproject-link wb-otherproject-wikibooks mw-list-item"><a href="https://id.wikibooks.org/wiki/Kalkulus" hreflang="id"><span>Wikibuku</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q149972" title="Pranala untuk menghubungkan butir pada ruang penyimpanan data [g]" accesskey="g"><span>Butir di Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav 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3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar sidebar-collapse plainlist"><tbody><tr><th class="sidebar-title" style="padding-bottom:0.25em;"><a class="mw-selflink selflink">Kalkulus</a></th></tr><tr><td class="sidebar-above" style="padding:0.15em 0.25em 0.3em;font-weight:normal;"> <ul><li><a href="/wiki/Teorema_dasar_kalkulus" title="Teorema dasar kalkulus">Teorema dasar</a></li></ul> <div class="hlist" style="margin-left: 0em;"> <ul><li><a href="/wiki/Limit_fungsi" title="Limit fungsi">Limit fungsi</a></li> <li><a href="/wiki/Fungsi_kontinu" title="Fungsi kontinu">Kontinuitas</a></li></ul> </div><div class="hlist" style="margin-left: 0em;"> <ul><li><a href="/wiki/Teorema_nilai_purata" title="Teorema nilai purata">Teorema nilai purata</a></li> <li><a href="/wiki/Teorema_Rolle" title="Teorema Rolle">Teorema Rolle</a></li></ul> </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;;display:block;margin-top:0.65em;"><span style="font-size:110%;"><a href="/wiki/Kalkulus_diferensial" title="Kalkulus diferensial">Diferensial</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;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r26333518"><table class="sidebar" style="border-collapse: collapse; border-spacing: 0px; border:none; width:100%; margin:0px; font-size: 100%; clear:none; float:none;"><tbody><tr><th class="sidebar-heading"> Definisi</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Turunan" title="Turunan">Turunan</a> (<a href="/w/index.php?title=Perumuman_dari_turunan&action=edit&redlink=1" class="new" title="Perumuman dari turunan (halaman belum tersedia)">perumuman</a>)</li></ul> <div class="hlist" style="margin-left: 0em; padding:0.1em 0;line-height:1.2em;"> <ul><li><a href="/wiki/Tabel_turunan" title="Tabel turunan">Tabel turunan</a></li> <li><a href="/wiki/Diferensial_(matematika)" class="mw-disambig" title="Diferensial (matematika)">Diferensial</a> <ul><li><a href="/w/index.php?title=Diferensial_(infinitesimal)&action=edit&redlink=1" class="new" title="Diferensial (infinitesimal) (halaman belum tersedia)">infinitesimal</a></li> <li><a href="/wiki/Turunan_fungsi" class="mw-redirect" title="Turunan fungsi">fungsi</a></li> <li><a href="/wiki/Diferensial_total" title="Diferensial total">total</a></li></ul></li></ul> </div></td> </tr><tr><th class="sidebar-heading"> Konsep</th></tr><tr><td class="sidebar-content hlist"> <div class="hlist" style="margin-left: 0em;"> <ul><li><a href="/w/index.php?title=Notasi_untuk_pendiferensialan&action=edit&redlink=1" class="new" title="Notasi untuk pendiferensialan (halaman belum tersedia)">Notasi untuk pendiferensialan</a></li> <li><a href="/wiki/Turunan_kedua" title="Turunan kedua">Turunan kedua</a></li> <li><a href="/wiki/Turunan_ketiga" class="mw-redirect" title="Turunan ketiga">Turunan ketiga</a></li> <li><a href="/w/index.php?title=Perubahan_variabel&action=edit&redlink=1" class="new" title="Perubahan variabel (halaman belum tersedia)">Perubahan variabel</a></li> <li><a href="/wiki/Fungsi_implisit" title="Fungsi implisit">Pendiferensialan implisit</a></li> <li><a href="/w/index.php?title=Laju_yang_berkaitan&action=edit&redlink=1" class="new" title="Laju yang berkaitan (halaman belum tersedia)">Laju yang berkaitan</a></li> <li><a href="/wiki/Teorema_Taylor" title="Teorema Taylor">Teorema Taylor</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Kaidah_diferensiasi" class="mw-redirect" title="Kaidah diferensiasi">Kaidah dan identitas</a></th></tr><tr><td class="sidebar-content hlist"> <div class="hlist" style="margin-left: 0em;"> <ul><li><a href="/w/index.php?title=Kaidah_penjumlahan_dalam_pendiferensialan&action=edit&redlink=1" class="new" title="Kaidah penjumlahan dalam pendiferensialan (halaman belum tersedia)">Kaidah penjumlahan dalam pendiferensialan</a></li> <li><a href="/wiki/Kaidah_darab" title="Kaidah darab">Perkalian</a></li> <li><a href="/wiki/Kaidah_rantai" title="Kaidah rantai">Rantai</a></li> <li><a href="/wiki/Kaidah_pangkat" title="Kaidah pangkat">Pangkat</a></li> <li><a href="/wiki/Kaidah_hasil-bagi" title="Kaidah hasil-bagi">Pembagian</a></li> <li><a href="/w/index.php?title=Rumus_Fa%C3%A0_di_Bruno&action=edit&redlink=1" class="new" title="Rumus Faà di Bruno (halaman belum tersedia)">Rumus Faà di Bruno</a></li></ul> </div></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;"><span style="font-size:110%;"><a href="/wiki/Integral" title="Integral">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;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r26333518"><table class="sidebar" style="border-collapse: collapse; border-spacing: 0px; border:none; width:100%; margin:0px; font-size: 100%; clear:none; float:none;"><tbody><tr><th class="sidebar-heading"> Definisi</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Antiderivatif" class="mw-redirect" title="Antiderivatif">Antiderivatif</a></li> <li><a href="/wiki/Integral" title="Integral">Integral</a> (<a href="/wiki/Integral_takwajar" title="Integral takwajar">takwajar</a>)</li> <li><a href="/wiki/Integral_Riemann" title="Integral Riemann">Integral Riemann</a></li> <li><a href="/wiki/Integrasi_Lebesgue-Stieltjes" title="Integrasi Lebesgue-Stieltjes">Integrasi Lebesgue</a></li> <li><a href="/w/index.php?title=Metode_integrasi_kontur&action=edit&redlink=1" class="new" title="Metode integrasi kontur (halaman belum tersedia)">Integrasi kontur</a></li> <li><a href="/wiki/Tabel_integral" title="Tabel integral">Tabel integral</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Integrasi secara</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Integrasi_parsial" title="Integrasi parsial">parsial</a></li> <li><a href="/wiki/Integrasi_cakram" title="Integrasi cakram">cakram</a></li> <li><a href="/wiki/Integrasi_kulit" title="Integrasi kulit">kulit tabung</a></li> <li><a href="/wiki/Integral_substitusi" title="Integral substitusi">substitusi</a> (<a href="/w/index.php?title=Substitusi_trigonometri&action=edit&redlink=1" class="new" title="Substitusi trigonometri (halaman belum tersedia)">trigonometri</a>)</li> <li><a href="/w/index.php?title=Pecahan_parsial_dalam_integrasi&action=edit&redlink=1" class="new" title="Pecahan parsial dalam integrasi (halaman belum tersedia)">pecahan parsial</a></li> <li><a href="/w/index.php?title=Urutan_integrasi_(kalkulus)&action=edit&redlink=1" class="new" title="Urutan integrasi (kalkulus) (halaman belum tersedia)">Urutan</a></li> <li><a href="/w/index.php?title=Integrasi_dengan_rumus_reduksi&action=edit&redlink=1" class="new" title="Integrasi dengan rumus reduksi (halaman belum tersedia)">Rumus reduksi</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;"><span style="font-size:110%;"><a href="/wiki/Deret_(matematika)" title="Deret (matematika)">Deret</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;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r26333518"><table class="sidebar" style="border-collapse: collapse; border-spacing: 0px; border:none; width:100%; margin:0px; font-size: 100%; clear:none; float:none;"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Deret_geometri" class="mw-redirect" title="Deret geometri">geometri</a> (<a href="/w/index.php?title=Deret_aritmetika-geometrik&action=edit&redlink=1" class="new" title="Deret aritmetika-geometrik (halaman belum tersedia)">aritmetika-geometrik</a>)</li> <li><a href="/wiki/Deret_harmonik_(matematika)" title="Deret harmonik (matematika)">harmonik</a></li> <li><a href="/wiki/Deret_selang-seling" title="Deret selang-seling">selang-seling</a></li> <li><a href="/wiki/Deret_pangkat" title="Deret pangkat">pangkat</a></li> <li><a href="/w/index.php?title=Deret_binomial&action=edit&redlink=1" class="new" title="Deret binomial (halaman belum tersedia)">binomial</a></li> <li><a href="/wiki/Deret_Taylor" title="Deret Taylor">Taylor</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Uji_kekonvergenan" title="Uji kekonvergenan">Uji kekonvergenan</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Uji_suku" title="Uji suku">uji suku</a></li> <li><a href="/w/index.php?title=Uji_rasio&action=edit&redlink=1" class="new" title="Uji rasio (halaman belum tersedia)">rasio</a></li> <li><a href="/w/index.php?title=Uji_akar&action=edit&redlink=1" class="new" title="Uji akar (halaman belum tersedia)">akar</a></li> <li><a href="/w/index.php?title=Uji_integral_untuk_kekonvergenan&action=edit&redlink=1" class="new" title="Uji integral untuk kekonvergenan (halaman belum tersedia)">integral</a></li> <li><a href="/w/index.php?title=Uji_perbandingan_langsung&action=edit&redlink=1" class="new" title="Uji perbandingan langsung (halaman belum tersedia)">perbandingan langsung</a></li> <li><br /><a href="/w/index.php?title=Uji_perbandingan_limit&action=edit&redlink=1" class="new" title="Uji perbandingan limit (halaman belum tersedia)">perbandingan limit</a></li> <li><a href="/w/index.php?title=Uji_deret_selang-seling&action=edit&redlink=1" class="new" title="Uji deret selang-seling (halaman belum tersedia)">deret selang-seling</a></li> <li><a href="/w/index.php?title=Uji_kondensasi_Cauchy&action=edit&redlink=1" class="new" title="Uji kondensasi Cauchy (halaman belum tersedia)">kondensasi Cauchy</a></li> <li><a href="/w/index.php?title=Uji_Dirichlet&action=edit&redlink=1" class="new" title="Uji Dirichlet (halaman belum tersedia)">Dirichlet</a></li> <li><a href="/w/index.php?title=Uji_Abel&action=edit&redlink=1" class="new" title="Uji Abel (halaman belum tersedia)">Abel</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;"><span style="font-size:110%;"><a href="/wiki/Kalkulus_vektor" title="Kalkulus vektor">Vektor</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;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r26333518"><table class="sidebar" style="border-collapse: collapse; border-spacing: 0px; border:none; width:100%; margin:0px; font-size: 100%; clear:none; float:none;"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Gradien" title="Gradien">Gradien</a></li> <li><a href="/wiki/Divergence" title="Divergence">Divergence</a></li> <li><a href="/w/index.php?title=Keikalan_(matematika)&action=edit&redlink=1" class="new" title="Keikalan (matematika) (halaman belum tersedia)">Keikalan</a></li> <li><a href="/w/index.php?title=Operator_Laplace&action=edit&redlink=1" class="new" title="Operator Laplace (halaman belum tersedia)">Laplace</a></li> <li><a href="/w/index.php?title=Turunan_berarah&action=edit&redlink=1" class="new" title="Turunan berarah (halaman belum tersedia)">berarah</a></li> <li><a href="/w/index.php?title=Identitas_kalkulus_vektor&action=edit&redlink=1" class="new" title="Identitas kalkulus vektor (halaman belum tersedia)">identitas</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Teorema</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/w/index.php?title=Teorema_kedivergenan&action=edit&redlink=1" class="new" title="Teorema kedivergenan (halaman belum tersedia)">Kedivergenan</a></li> <li><a href="/w/index.php?title=Teorema_gradien&action=edit&redlink=1" class="new" title="Teorema gradien (halaman belum tersedia)">Gradien</a></li> <li><a href="/wiki/Teorema_Green" title="Teorema Green">Green</a></li> <li><a href="/wiki/Teorema_Stokes" class="mw-redirect" title="Teorema Stokes">Stokes</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;"><span style="font-size:110%;"><a href="/wiki/Kalkulus_multivariabel" title="Kalkulus multivariabel">Multivariabel</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;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r26333518"><table class="sidebar" style="border-collapse: collapse; border-spacing: 0px; border:none; width:100%; margin:0px; font-size: 100%; clear:none; float:none;"><tbody><tr><th class="sidebar-heading"> Formalisme</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Kalkulus_matriks" title="Kalkulus matriks">matriks</a></li> <li><a href="/w/index.php?title=Kalkulus_tensor&action=edit&redlink=1" class="new" title="Kalkulus tensor (halaman belum tersedia)">tensor</a></li> <li><a href="/w/index.php?title=Turunan_luar&action=edit&redlink=1" class="new" title="Turunan luar (halaman belum tersedia)">eksterior</a></li> <li><a href="/w/index.php?title=Kalkulu_geometrik&action=edit&redlink=1" class="new" title="Kalkulu geometrik (halaman belum tersedia)">geometrik</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Definisi</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Turunan_parsial" title="Turunan parsial">Turunan parsial</a></li> <li><a href="/wiki/Integral_lipat" title="Integral lipat">Integral lipat</a></li> <li><a href="/wiki/Integral_garis" title="Integral garis">Integral garis</a></li> <li><a href="/wiki/Permukaan_integral" class="mw-redirect" title="Permukaan integral">Permukaan integral</a></li> <li><a href="/wiki/Integral_volume" title="Integral volume">integral volume</a></li> <li><a href="/wiki/Matriks_dan_determinan_Jacobi" class="mw-redirect" title="Matriks dan determinan Jacobi">Jacobi</a></li> <li><a href="/wiki/Matriks_Hesse" title="Matriks Hesse">Hesse</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;"><span style="font-size:110%;">Khusus</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;"><div class="hlist" style="margin-left: 0em;"> <ul><li><a href="/w/index.php?title=Kalkulus_fraksional&action=edit&redlink=1" class="new" title="Kalkulus fraksional (halaman belum tersedia)">fraksional</a></li> <li><a href="/w/index.php?title=Kalkulus_Malliavin&action=edit&redlink=1" class="new" title="Kalkulus Malliavin (halaman belum tersedia)">Malliavin</a></li> <li><a href="/w/index.php?title=Kalkulus_stokastik&action=edit&redlink=1" class="new" title="Kalkulus stokastik (halaman belum tersedia)">stokastik</a></li> <li><a href="/w/index.php?title=Kalkulus_variasi&action=edit&redlink=1" class="new" title="Kalkulus variasi (halaman belum tersedia)">variasi</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-navbar"><style data-mw-deduplicate="TemplateStyles:r18590415">.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-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}.mw-parser-output .infobox .navbar{font-size:100%}.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}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-lihat"><a href="/wiki/Templat:Kalkulus" title="Templat:Kalkulus"><abbr title="Lihat templat ini">l</abbr></a></li><li class="nv-bicara"><a href="/wiki/Pembicaraan_Templat:Kalkulus" title="Pembicaraan Templat:Kalkulus"><abbr title="Diskusikan templat ini">b</abbr></a></li><li class="nv-sunting"><a class="external text" href="https://id.wikipedia.org/w/index.php?title=Templat:Kalkulus&action=edit"><abbr title="Sunting templat ini">s</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Kalkulus</b> (<a href="/wiki/Bahasa_Latin" title="Bahasa Latin">bahasa Latin</a>: <span lang="la"><i>calculus</i></span>, artinya "batu kecil", untuk menghitung) adalah cabang ilmu <a href="/wiki/Matematika" title="Matematika">matematika</a> yang mencakup <a href="/wiki/Limit_(matematika)" title="Limit (matematika)">limit</a>, <a href="/wiki/Turunan" title="Turunan">turunan</a>, <a href="/wiki/Integral" title="Integral">integral</a>, dan <a href="/wiki/0,999...#Deret_dan_barisan_takterhingga" title="0,999...">deret takterhingga</a>. Kalkulus adalah ilmu yang mempelajari perubahan, sebagaimana <a href="/wiki/Geometri" title="Geometri">geometri</a> yang mempelajari bentuk dan <a href="/wiki/Aljabar" title="Aljabar">aljabar</a> yang mempelajari operasi dan penerapannya untuk memecahkan persamaan. Kalkulus memiliki aplikasi yang luas dalam bidang-bidang <a href="/wiki/Ilmu" title="Ilmu">sains</a>, <a href="/wiki/Ekonomi" title="Ekonomi">ekonomi</a>, dan <a href="/wiki/Teknik" class="mw-redirect" title="Teknik">teknik</a>; serta dapat memecahkan berbagai masalah yang tidak dapat dipecahkan dengan <a href="/wiki/Aljabar_elementer" title="Aljabar elementer">aljabar elementer</a>.<sup id="cite_ref-concepts_1-0" class="reference"><a href="#cite_note-concepts-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Kalkulus memiliki dua cabang utama, <b><a href="/wiki/Kalkulus_diferensial" title="Kalkulus diferensial">kalkulus diferensial</a></b> dan <b><a href="/wiki/Integral" title="Integral">kalkulus integral</a></b> yang saling berhubungan melalui <a href="/wiki/Teorema_dasar_kalkulus" title="Teorema dasar kalkulus">teorema dasar kalkulus</a>. Contoh cabang kalkulus yang lain adalah kalkulus proposisional, kalkulus variasi, kalkulus lambda, dan kalkulus proses. Pelajaran kalkulus adalah pintu gerbang menuju pelajaran matematika lainnya yang lebih tinggi, yang khusus mempelajari <a href="/wiki/Fungsi_(matematika)" title="Fungsi (matematika)">fungsi</a> dan <a href="/wiki/Limit" class="mw-disambig" title="Limit">limit</a>, yang secara umum dinamakan <a href="/wiki/Analisis_matematika" class="mw-redirect" title="Analisis matematika">analisis matematika</a>.<sup id="cite_ref-concepts_1-1" class="reference"><a href="#cite_note-concepts-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Sejarah">Sejarah</h2></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Berkas:GodfreyKneller-IsaacNewton-1689.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/GodfreyKneller-IsaacNewton-1689.jpg/200px-GodfreyKneller-IsaacNewton-1689.jpg" decoding="async" width="200" height="281" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/GodfreyKneller-IsaacNewton-1689.jpg/300px-GodfreyKneller-IsaacNewton-1689.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/39/GodfreyKneller-IsaacNewton-1689.jpg/400px-GodfreyKneller-IsaacNewton-1689.jpg 2x" data-file-width="1364" data-file-height="1916" /></a><figcaption><i><a href="/wiki/Sir_Isaac_Newton" class="mw-redirect" title="Sir Isaac Newton">Sir Isaac Newton</a></i> adalah salah seorang penemu dan kontributor kalkulus yang terkenal.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Perkembangan">Perkembangan</h3></div> <p>Sejarah perkembangan kalkulus bisa ditilik pada beberapa periode zaman, yaitu <a href="/wiki/Abad_Kuno" class="mw-redirect" title="Abad Kuno">zaman kuno</a>, <a href="/wiki/Abad_Pertengahan" title="Abad Pertengahan">zaman pertengahan</a>, dan <a href="/wiki/Zaman_modern" title="Zaman modern">zaman modern</a>. Pada periode zaman kuno, beberapa pemikiran tentang kalkulus integral telah muncul, tetapi tidak dikembangkan dengan baik dan sistematis.<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> Perhitungan <a href="/wiki/Volume" title="Volume">volume</a> dan luas yang merupakan fungsi utama dari kalkulus integral bisa ditelusuri kembali pada <a href="/wiki/Papirus_Matematika_Moskwa" class="mw-redirect" title="Papirus Matematika Moskwa">Papirus Moskwa</a> <a href="/wiki/Mesir" title="Mesir">Mesir</a> (c. 1800 SM). Pada papirus tersebut, orang Mesir telah mampu menghitung volume <a href="/wiki/Piramid" class="mw-redirect" title="Piramid">piramida</a> terpancung.<sup id="cite_ref-Aslaksen_3-0" class="reference"><a href="#cite_note-Aslaksen-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> mengembangkan pemikiran ini lebih jauh dan menciptakan <a href="/wiki/Heuristik" class="mw-redirect" title="Heuristik">heuristik</a> yang menyerupai <a href="/wiki/Integral" title="Integral">kalkulus integral</a>.<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> </p><p>Pada zaman pertengahan, matematikawan <a href="/wiki/India" title="India">India</a>, <a href="/wiki/Aryabhata" title="Aryabhata">Aryabhata</a>, menggunakan konsep kecil tak terhingga pada tahun <a href="/wiki/499" title="499">499</a> dan mengekspresikan masalah astronomi dalam bentuk <a href="/wiki/Persamaan_diferensial" title="Persamaan diferensial">persamaan diferensial</a> dasar.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Persamaan ini kemudian mengantar <a href="/wiki/Bh%C4%81skara_II" title="Bhāskara II">Bhāskara II</a> pada abad ke-12 untuk mengembangkan bentuk awal <a href="/wiki/Turunan" title="Turunan">turunan</a> yang mewakili perubahan yang sangat kecil takterhingga dan menjelaskan bentuk awal dari "<a href="/wiki/Teorema_Rolle" title="Teorema Rolle">Teorema Rolle</a>".<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Sekitar tahun <a href="/wiki/1000" title="1000">1000</a>, matematikawan <a href="/wiki/Irak" title="Irak">Irak</a> <a href="/wiki/Ibnu_Haitham" class="mw-redirect" title="Ibnu Haitham">Ibn al-Haytham</a> (Alhazen) menjadi orang pertama yang menurunkan rumus perhitungan hasil jumlah pangkat empat, dan dengan menggunakan <a href="/wiki/Induksi_matematika" title="Induksi matematika">induksi matematika</a>, dia mengembangkan suatu metode untuk menurunkan rumus umum dari hasil pangkat integral yang sangat penting terhadap perkembangan kalkulus integral.<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> Pada abad ke-12, seorang <a href="/wiki/Persia" class="mw-redirect" title="Persia">Persia</a> <a href="/wiki/Sharaf_al-Din_al-Tusi" title="Sharaf al-Din al-Tusi">Sharaf al-Din al-Tusi</a> menemukan <a href="/wiki/Turunan" title="Turunan">turunan</a> dari <a href="/wiki/Fungsi_kubik" title="Fungsi kubik">fungsi kubik</a>, sebuah hasil yang penting dalam kalkulus diferensial.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Pada abad ke-14, <a href="/wiki/Madhava_dari_Sangamagrama" title="Madhava dari Sangamagrama">Madhava</a>, bersama dengan matematikawan-astronom dari <a href="/wiki/Mazhab_astronomi_dan_matematika_Kerala" title="Mazhab astronomi dan matematika Kerala">mazhab astronomi dan matematika Kerala</a>, menjelaskan kasus khusus dari <a href="/wiki/Deret_Taylor" title="Deret Taylor">deret Taylor</a>,<sup id="cite_ref-madhava_9-0" class="reference"><a href="#cite_note-madhava-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> yang dituliskan dalam teks <i><a href="/wiki/Yuktibhasa" title="Yuktibhasa">Yuktibhasa</a></i>.<sup id="cite_ref-scotlnd_10-0" class="reference"><a href="#cite_note-scotlnd-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-pdffile3_11-0" class="reference"><a href="#cite_note-pdffile3-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-charles_12-0" class="reference"><a href="#cite_note-charles-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>Pada zaman modern, penemuan independen terjadi pada awal abad ke-17 di Jepang oleh matematikawan seperti <a href="/wiki/Seki_K%C5%8Dwa" title="Seki Kōwa">Seki Kowa</a>. Di Eropa, beberapa matematikawan seperti <a href="/wiki/John_Wallis" title="John Wallis">John Wallis</a> dan <a href="/wiki/Isaac_Barrow" title="Isaac Barrow">Isaac Barrow</a> memberikan terobosan dalam kalkulus.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> <a href="/wiki/James_Gregory" title="James Gregory">James Gregory</a> membuktikan sebuah kasus khusus dari <a href="/wiki/Teorema_dasar_kalkulus" title="Teorema dasar kalkulus">teorema dasar kalkulus</a> pada tahun 1668.<sup id="cite_ref-Simmons_14-0" class="reference"><a href="#cite_note-Simmons-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Gottfried_Wilhelm_von_Leibniz.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Gottfried_Wilhelm_von_Leibniz.jpg/200px-Gottfried_Wilhelm_von_Leibniz.jpg" decoding="async" width="200" height="253" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Gottfried_Wilhelm_von_Leibniz.jpg/300px-Gottfried_Wilhelm_von_Leibniz.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/6/6a/Gottfried_Wilhelm_von_Leibniz.jpg 2x" data-file-width="316" data-file-height="400" /></a><figcaption><i><a href="/wiki/Gottfried_Leibniz" title="Gottfried Leibniz">Gottfried Wilhelm Leibniz</a></i> pada awalnya dituduh menjiplak dari hasil kerja Sir Isaac Newton yang tidak dipublikasikan, namun sekarang dianggap sebagai kontributor kalkulus yang hasil kerjanya dilakukan secara terpisah.</figcaption></figure> <p><a href="/wiki/Gottfried_Leibniz" title="Gottfried Leibniz">Leibniz</a> dan <a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a> mendorong pemikiran-pemikiran ini bersama sebagai sebuah kesatuan dan kedua orang ilmuwan tersebut dianggap sebagai penemu kalkulus secara terpisah dalam waktu yang hampir bersamaan. Newton mengaplikasikan kalkulus secara umum ke bidang <a href="/wiki/Fisika" title="Fisika">fisika</a> sementara Leibniz mengembangkan notasi-notasi kalkulus yang banyak digunakan sekarang.<sup id="cite_ref-Simmons_14-1" class="reference"><a href="#cite_note-Simmons-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>Ketika Newton dan Leibniz mempublikasikan hasil mereka untuk pertama kali, timbul kontroversi di antara matematikawan tentang mana yang lebih pantas untuk menerima penghargaan terhadap kerja mereka. Newton menurunkan hasil kerjanya terlebih dahulu, tetapi Leibniz yang pertama kali mempublikasikannya. Newton menuduh Leibniz mencuri pemikirannya dari catatan-catatan yang tidak dipublikasikan, yang sering dipinjamkan Newton kepada beberapa anggota dari <a href="/wiki/Royal_Society" title="Royal Society">Royal Society</a>.<sup id="cite_ref-leibniz_15-0" class="reference"><a href="#cite_note-leibniz-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>Pemeriksaan secara terperinci menunjukkan bahwa keduanya bekerja secara terpisah, dengan Leibniz memulai dari integral dan Newton dari turunan. Sekarang, baik Newton dan Leibniz diberikan penghargaan dalam mengembangkan kalkulus secara terpisah. Adalah Leibniz yang memberikan nama kepada ilmu cabang matematika ini sebagai kalkulus, sedangkan Newton menamakannya "The science of fluxions".<sup id="cite_ref-leibniz_15-1" class="reference"><a href="#cite_note-leibniz-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>Sejak itu, banyak matematikawan yang memberikan kontribusi terhadap pengembangan lebih lanjut dari kalkulus. Salah satu karya perdana yang paling lengkap mengenai analisis finit dan infinitesimal ditulis pada tahun 1748 oleh <a href="/wiki/Maria_Gaetana_Agnesi" title="Maria Gaetana Agnesi">Maria Gaetana Agnesi</a>.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Maria_Gaetana_Agnesi.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Maria_Gaetana_Agnesi.jpg/150px-Maria_Gaetana_Agnesi.jpg" decoding="async" width="150" height="174" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Maria_Gaetana_Agnesi.jpg/225px-Maria_Gaetana_Agnesi.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/57/Maria_Gaetana_Agnesi.jpg/300px-Maria_Gaetana_Agnesi.jpg 2x" data-file-width="344" data-file-height="400" /></a><figcaption><a href="/wiki/Maria_Gaetana_Agnesi" title="Maria Gaetana Agnesi">Maria Gaetana Agnesi</a></figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Pengaruh_penting">Pengaruh penting</h3></div> <p>Walau beberapa konsep kalkulus telah dikembangkan terlebih dahulu di Mesir, Yunani, Tiongkok, India, Iraq, Persia, dan Jepang, penggunaaan kalkulus modern dimulai di <a href="/wiki/Eropa" title="Eropa">Eropa</a> pada abad ke-17 sewaktu <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> dan <a href="/wiki/Gottfried_Leibniz" title="Gottfried Leibniz">Gottfried Wilhelm Leibniz</a> mengembangkan prinsip dasar kalkulus. Hasil kerja mereka kemudian memberikan pengaruh yang kuat terhadap perkembangan <a href="/wiki/Fisika" title="Fisika">fisika</a>.<sup id="cite_ref-Simmons_14-2" class="reference"><a href="#cite_note-Simmons-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>Aplikasi kalkulus diferensial meliputi perhitungan <a href="/wiki/Kecepatan" title="Kecepatan">kecepatan</a> dan <a href="/wiki/Percepatan" title="Percepatan">percepatan</a>, <a href="/wiki/Gradien" title="Gradien">kemiringan</a> suatu kurva, dan optimalisasi. Aplikasi dari kalkulus integral meliputi perhitungan <a href="/wiki/Luas" title="Luas">luas</a>, <a href="/wiki/Volume" title="Volume">volume</a>, panjang busur, <a href="/wiki/Pusat_massa" title="Pusat massa">pusat massa</a>, <a href="/wiki/Kerja" class="mw-disambig" title="Kerja">kerja</a>, dan <a href="/wiki/Tekanan" title="Tekanan">tekanan</a>. Aplikasi lebih jauh meliputi <a href="/wiki/Deret_pangkat" title="Deret pangkat">deret pangkat</a> dan <a href="/wiki/Deret_Fourier" title="Deret Fourier">deret Fourier</a>.<sup id="cite_ref-Simmons_14-3" class="reference"><a href="#cite_note-Simmons-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>Kalkulus juga digunakan untuk mendapatkan pemahaman yang lebih rinci mengenai ruang, waktu, dan gerak. Selama berabad-abad, para matematikawan dan filsuf berusaha memecahkan paradoks yang meliputi pembagian bilangan dengan nol ataupun jumlah dari deret takterhingga. Seorang filsuf Yunani kuno memberikan beberapa contoh terkenal seperti <a href="/wiki/Paradoks_Zeno" title="Paradoks Zeno">paradoks Zeno</a>. Kalkulus memberikan solusi, terutama di bidang limit dan deret takterhingga, yang kemudian berhasil memecahkan paradoks tersebut.<sup id="cite_ref-Simmons_14-4" class="reference"><a href="#cite_note-Simmons-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Prinsip_dasar">Prinsip dasar</h2></div> <style data-mw-deduplicate="TemplateStyles:r18844875">.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}</style><div role="note" class="hatnote navigation-not-searchable">Lihat pula: <a href="/wiki/Daftar_topik_kalkulus" title="Daftar topik kalkulus">Daftar topik kalkulus</a></div> <div class="mw-heading mw-heading3"><h3 id="Limit_dan_kecil_tak_terhingga">Limit dan kecil tak terhingga</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18844875"><div role="note" class="hatnote navigation-not-searchable">Artikel utama: <a href="/wiki/Limit_(matematika)" title="Limit (matematika)">Limit (matematika)</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Berkas:L%C3%ADmite_01.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/L%C3%ADmite_01.svg/300px-L%C3%ADmite_01.svg.png" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/L%C3%ADmite_01.svg/450px-L%C3%ADmite_01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d1/L%C3%ADmite_01.svg/600px-L%C3%ADmite_01.svg.png 2x" data-file-width="800" data-file-height="800" /></a><figcaption>Definisi limit mengatakan bahwa ketika <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> mendekati titik <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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>, maka limit <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> mendekati <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span>, jika untuk setiap bilangan <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>, terdapat bilangan <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> sedemikian rupa sehingga <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 0<|x-p|<\delta \Longrightarrow |f(x)-L|<\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>−</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>δ</mi> <mo stretchy="false">⟹</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> <mo>−</mo> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<|x-p|<\delta \Longrightarrow |f(x)-L|<\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34bada1060e090728136ab40996dae7e5681c8c9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.454ex; height:2.843ex;" alt="{\displaystyle 0<|x-p|<\delta \Longrightarrow |f(x)-L|<\varepsilon }"></span></figcaption></figure> <p>Kalkulus pada umumnya dikembangkan dengan memanipulasi sejumlah kuantitas yang sangat kecil. Objek ini, yang dapat diperlakukan sebagai angka, adalah sangat kecil. Sebuah bilangan <i>dx</i> yang kecilnya tak terhingga dapat lebih besar daripada 0, namun lebih kecil daripada bilangan apapun pada deret 1, <span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; margin:0 0.1em;">1</span><span style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">2</span></span>, <span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; margin:0 0.1em;">1</span><span style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">3</span></span>, ... dan bilangan real positif apapun. Setiap perkalian dengan kecil tak terhingga (infinitesimal) tetaplah kecil tak terhingga, dengan kata lain kecil tak terhingga tidak memenuhi "ciri-ciri Archimedes". Dari sudut pandang ini, kalkulus adalah sekumpulan teknik untuk memanipulasi kecil tak terhingga.<sup id="cite_ref-Larson_17-0" class="reference"><a href="#cite_note-Larson-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>Pada abad ke-19, konsep kecil tak terhingga ini ditinggalkan karena tidak cukup cermat, sebaliknya ia digantikan oleh konsep <a href="/wiki/Limit_(matematika)" title="Limit (matematika)">limit (matematika)</a>. Limit menjelaskan nilai suatu fungsi pada nilai input tertentu dengan hasil dari nilai input terdekat. Dari sudut pandang ini, kalkulus adalah sekumpulan teknik memanipulasi limit-limit tertentu.<sup id="cite_ref-Larson_17-1" class="reference"><a href="#cite_note-Larson-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> Secara cermat, definisi limit suatu fungsi adalah: </p> <style data-mw-deduplicate="TemplateStyles:r21477316">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 40px}.mw-parser-output .templatequote .templatequotecite{line-height:1.5em;text-align:left;padding-left:1.6em;margin-top:0}</style><blockquote class="templatequote"><p>Diberikan fungsi <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> yang didefinisikan pada interval di <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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>, terkecuali mungkin pada <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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> itu sendiri. Ketika <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> mendekati <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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>, maka limit <i><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></i> dapat dikatakan mendekati <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span>, dan dituliskan sebagai: </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 \lim _{x\to p}{f(x)}=L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>p</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to p}{f(x)}=L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccb6be9d1db1cfecdf046657f4bf7c0971ead85f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.896ex; height:4.176ex;" alt="{\displaystyle \lim _{x\to p}{f(x)}=L}"></span></dd></dl> <p>jika, untuk setiap bilangan <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>, terdapat bilangan <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> yang berkoresponden dengannya sedemikian rupa untuk setiap <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></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 0<|x-p|<\delta \Longrightarrow |f(x)-L|<\varepsilon \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>−</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>δ</mi> <mo stretchy="false">⟹</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> <mo>−</mo> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ε</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<|x-p|<\delta \Longrightarrow |f(x)-L|<\varepsilon \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c07f8475c147ae5e45f6c9daa0061c1d5e1369a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.841ex; height:2.843ex;" alt="{\displaystyle 0<|x-p|<\delta \Longrightarrow |f(x)-L|<\varepsilon \,}"></span></dd></dl></blockquote> <div class="mw-heading mw-heading3"><h3 id="Turunan">Turunan</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18844875"><div role="note" class="hatnote navigation-not-searchable">Artikel utama: <a href="/wiki/Turunan" title="Turunan">Turunan</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Derivative.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Derivative.svg/250px-Derivative.svg.png" decoding="async" width="250" height="277" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Derivative.svg/375px-Derivative.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/18/Derivative.svg/500px-Derivative.svg.png 2x" data-file-width="1158" data-file-height="1282" /></a><figcaption>Grafik fungsi turunan.</figcaption></figure> <p>Turunan dari suatu fungsi mewakili perubahan yang sangat kecil dari fungsi tersebut terhadap variabelnya. Proses menemukan turunan dari suatu fungsi disebut sebagai pendiferensialan ataupun diferensiasi.<sup id="cite_ref-concepts_1-2" class="reference"><a href="#cite_note-concepts-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Secara matematis, turunan fungsi <i><b><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></b></i> terhadap variabel <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> adalah <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'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/258eaada38956fb69b8cb1a2eef46bcb97d3126b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.005ex; height:2.843ex;" alt="{\displaystyle f'}"></span> yang nilainya pada titik <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> adalah: </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)=\lim _{h\to 0}{f(x+h)-f(x) \over {h}}}"> <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> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'(x)=\lim _{h\to 0}{f(x+h)-f(x) \over {h}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5775719bbbf17def42efd41bc92d81edf13b95f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.733ex; height:5.843ex;" alt="{\displaystyle f'(x)=\lim _{h\to 0}{f(x+h)-f(x) \over {h}}}"></span>,</dd></dl> <p>dengan syarat limit tersebut ada. Jika <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'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/258eaada38956fb69b8cb1a2eef46bcb97d3126b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.005ex; height:2.843ex;" alt="{\displaystyle f'}"></span> ada pada titik <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> tertentu, maka <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'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/258eaada38956fb69b8cb1a2eef46bcb97d3126b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.005ex; height:2.843ex;" alt="{\displaystyle f'}"></span> dapat dikatakan terdiferensialkan (memiliki turunan) pada <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>, dan jika <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'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/258eaada38956fb69b8cb1a2eef46bcb97d3126b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.005ex; height:2.843ex;" alt="{\displaystyle f'}"></span> ada di setiap titik pada domain <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>, maka <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> dapat disebut terdiferensialkan. </p><p>Jika <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=x+h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=x+h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b4668ee124b580b6cf41f119d096ee0d23872ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.696ex; height:2.343ex;" alt="{\displaystyle z=x+h}"></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="{\displaystyle h=z-x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <mi>z</mi> <mo>−</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h=z-x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a62377542af1887b5390f3a1dc6c514e11e4a850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.696ex; height:2.343ex;" alt="{\displaystyle h=z-x}"></span>, dan <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 h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> mendekati 0 <i>jika dan hanya jika</i> <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> mendekati <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>, maka definisi turunan di atas dapat ditulis pula sebagai: </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)=\lim _{z\to x}{f(z)-f(x) \over {z-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> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo stretchy="false">→</mo> <mi>x</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>−</mo> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'(x)=\lim _{z\to x}{f(z)-f(x) \over {z-x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b5c99317a4f20fdd0e701c5e7c20fd46dab61a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:24.253ex; height:5.843ex;" alt="{\displaystyle f'(x)=\lim _{z\to x}{f(z)-f(x) \over {z-x}}}"></span></dd></dl> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Tangent_derivative_calculusdia.jpeg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/id/thumb/1/19/Tangent_derivative_calculusdia.jpeg/250px-Tangent_derivative_calculusdia.jpeg" decoding="async" width="250" height="172" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/id/thumb/1/19/Tangent_derivative_calculusdia.jpeg/375px-Tangent_derivative_calculusdia.jpeg 1.5x, //upload.wikimedia.org/wikipedia/id/thumb/1/19/Tangent_derivative_calculusdia.jpeg/500px-Tangent_derivative_calculusdia.jpeg 2x" data-file-width="561" data-file-height="387" /></a><figcaption>Garis singgung pada <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,f(x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,f(x))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b21dd0c5c5815bc0516f679f631fd588ceb458d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.59ex; height:2.843ex;" alt="{\displaystyle (x,f(x))}"></span>. Turunan sebuah kurva <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"> <msup> <mi>f</mi> <mo>′</mo> </msup> <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/d0cd7d7c75340e779d82658e19d1720ce84ab127" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.144ex; height:3.009ex;" alt="{\displaystyle f'(x)}"></span> pada sebuah titik adalah kemiringan dari garis singgung yang menyinggung kurva pada titik tersebut.</figcaption></figure> <p>Perhatikan bahwa ekspresi <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+h)-f(x) \over {h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {f(x+h)-f(x) \over {h}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a434a000f3429aa25aa695f74c719715791e9d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.691ex; height:5.843ex;" alt="{\displaystyle {f(x+h)-f(x) \over {h}}}"></span> pada definisi turunan di atas merupakan gradien dari garis sekan yang melewati titik <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,f(x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,f(x))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b21dd0c5c5815bc0516f679f631fd588ceb458d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.59ex; height:2.843ex;" alt="{\displaystyle (x,f(x))}"></span> dan <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+h,f(x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+h,f(x))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9190b73f030b31bc42bad1156b2f7a80704888f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.77ex; height:2.843ex;" alt="{\displaystyle (x+h,f(x))}"></span> pada kurva <i><b><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></b></i>. Ketika limit <i><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 h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span></i> mendekati 0, maka kemiringan dari garis singgung yang diperoleh menyinggung kurva <i><b><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></b></i> pada titik <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>. Hal ini berarti pula garis singgung suatu kurva merupakan limit dari garis sekan, demikian pulanya turunan dari suatu fungsi <i><b><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></b></i> merupakan gradien dari fungsi tersebut.<sup id="cite_ref-concepts_1-3" class="reference"><a href="#cite_note-concepts-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Sebagai contoh, untuk menemukan gradien dari fungsi <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)=x^{2}}"> <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>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84ddac4ae10b1aa4a11741c79771a583419fb1fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.9ex; height:3.176ex;" alt="{\displaystyle f(x)=x^{2}}"></span> pada titik (3,9): </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}f'(3)&=\lim _{h\to 0}{(3+h)^{2}-9 \over {h}}\\&=\lim _{h\to 0}{9+6h+h^{2}-9 \over {h}}\\&=\lim _{h\to 0}{6h+h^{2} \over {h}}\\&=\lim _{h\to 0}(6+h)\\&=6\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> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo>+</mo> <mi>h</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>9</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>9</mn> <mo>+</mo> <mn>6</mn> <mi>h</mi> <mo>+</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>9</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>6</mn> <mi>h</mi> <mo>+</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mo stretchy="false">(</mo> <mn>6</mn> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>6</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f'(3)&=\lim _{h\to 0}{(3+h)^{2}-9 \over {h}}\\&=\lim _{h\to 0}{9+6h+h^{2}-9 \over {h}}\\&=\lim _{h\to 0}{6h+h^{2} \over {h}}\\&=\lim _{h\to 0}(6+h)\\&=6\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41dc47a004d522c397f1f2a4de1e31f21bd101db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.171ex; width:29.203ex; height:25.509ex;" alt="{\displaystyle {\begin{aligned}f'(3)&=\lim _{h\to 0}{(3+h)^{2}-9 \over {h}}\\&=\lim _{h\to 0}{9+6h+h^{2}-9 \over {h}}\\&=\lim _{h\to 0}{6h+h^{2} \over {h}}\\&=\lim _{h\to 0}(6+h)\\&=6\end{aligned}}}"></span></dd></dl> <p>Ilmu yang mempelajari definisi, sifat, dan aplikasi dari <a href="/wiki/Turunan" title="Turunan">turunan</a> atau <a href="/wiki/Gradien" title="Gradien">kemiringan</a> dari sebuah grafik disebut <a href="/wiki/Kalkulus_diferensial" title="Kalkulus diferensial">kalkulus diferensial</a> </p> <figure typeof="mw:File/Thumb"><a href="/wiki/Berkas:Sec2tan.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Sec2tan.gif/250px-Sec2tan.gif" decoding="async" width="250" height="241" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Sec2tan.gif/375px-Sec2tan.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/34/Sec2tan.gif/500px-Sec2tan.gif 2x" data-file-width="586" data-file-height="565" /></a><figcaption>Garis singgung sebagai limit dari garis sekan. Turunan dari kurva <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> di suatu titik adalah kemiringan dari garis singgung yang menyinggung kurva pada titik tersebut. Kemiringan ini ditentukan dengan memakai nilai limit dari kemiringan garis sekan.</figcaption></figure> <div class="mw-heading mw-heading4"><h4 id="Notasi_pendiferensialan">Notasi pendiferensialan</h4></div> <p>Terdapat berbagai macam notasi matematika yang dapat digunakan untuk menyatakan turunan, meliputi <a href="/wiki/Notasi_Leibniz" title="Notasi Leibniz">notasi Leibniz</a>, notasi Lagrange, <a href="/wiki/Notasi_Newton" title="Notasi Newton">notasi Newton</a>, dan notasi <a href="/wiki/Euler" class="mw-redirect" title="Euler">Euler</a>.<sup id="cite_ref-concepts_1-4" class="reference"><a href="#cite_note-concepts-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p><b>Notasi Leibniz</b> diperkenalkan oleh <a href="/wiki/Gottfried_Leibniz" title="Gottfried Leibniz">Gottfried Leibniz</a> dan merupakan salah satu notasi yang paling awal digunakan. Ia sering digunakan terutama ketika hubungan antar <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=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2311a6a75c54b0ea085a381ba472c31d59321514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.672ex; height:2.843ex;" alt="{\displaystyle y=f(x)}"></span> dipandang sebagai hubungan fungsional antara variabel bebas dengan variabel terikat. Turunan dari fungsi tersebut terhadap <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> ditulis sebagai:<sup id="cite_ref-leibniz_15-2" class="reference"><a href="#cite_note-leibniz-15"><span class="cite-bracket">[</span>15<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 {\frac {dy}{dx}},\quad {\frac {df}{dx}}(x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>f</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dy}{dx}},\quad {\frac {df}{dx}}(x),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5509cdcbc56c595edd8788ebf14d622740bb755b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.906ex; height:5.509ex;" alt="{\displaystyle {\frac {dy}{dx}},\quad {\frac {df}{dx}}(x),}"></span> ataupun <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 {\frac {d}{dx}}f(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}f(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17937a7b22b030d44dad85d56a9c3c9ad41c07ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.446ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dx}}f(x).}"></span></dd></dl> <p><b>Notasi Lagrange</b> diperkenalkan oleh <a href="/wiki/Joseph-Louis_de_Lagrange" title="Joseph-Louis de Lagrange">Joseph Louis Lagrange</a> dan merupakan notasi yang paling sering digunakan. Dalam notasi ini, turunan fungsi <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> ditulis sebagai <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"> <msup> <mi>f</mi> <mo>′</mo> </msup> <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/d0cd7d7c75340e779d82658e19d1720ce84ab127" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.144ex; height:3.009ex;" alt="{\displaystyle f'(x)}"></span> ataupun hanya <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'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/258eaada38956fb69b8cb1a2eef46bcb97d3126b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.005ex; height:2.843ex;" alt="{\displaystyle f'}"></span>. </p><p><b>Notasi Newton</b>, juga disebut sebagai notasi titik, menempatkan titik di atas fungsi untuk menandakan turunan. Jika <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=f(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=f(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcb3a22359174d9da835b238b25e2f89ce1cd2ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.181ex; height:2.843ex;" alt="{\displaystyle y=f(t)}"></span>, maka <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 {\dot {y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea068ce646833369cccf19795f23613159b5f89f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.302ex; height:2.509ex;" alt="{\displaystyle {\dot {y}}}"></span> mewakili turunan <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> terhadap <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 t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>. Notasi ini hampir secara eksklusif digunakan untuk melambangkan turunan terhadap waktu. Notasi ini sering terlihat dalam bidang <a href="/wiki/Fisika" title="Fisika">fisika</a> dan bidang matematika yang berhubungan dengan fisika. </p><p><b>Notasi <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a></b> menggunakan operator diferensial <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 D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> yang diterapkan pada fungsi <i><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span></i> untuk memberikan turunan pertamanya <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 Df}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Df}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/683279b8feba655b049a1c55f4bd9026218fcdb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.203ex; height:2.509ex;" alt="{\displaystyle Df}"></span>. Jika <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=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2311a6a75c54b0ea085a381ba472c31d59321514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.672ex; height:2.843ex;" alt="{\displaystyle y=f(x)}"></span> adalah variabel terikat, maka <i><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span></i> seringkali dilekatkan pada <i><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span></i> untuk mengklarifikasikan keterbebasan variabel <i><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span></i> Notasi Euler kemudian ditulis sebagai: </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 D_{x}y\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>y</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{x}y\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24105b495826cf50d3ed22987b51fa4f3cf364bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.639ex; height:2.509ex;" alt="{\displaystyle D_{x}y\,}"></span> atau <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 D_{x}f(x)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{x}f(x)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3728d37766091b7849295bcc2ac2c895eaef0b0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.901ex; height:2.843ex;" alt="{\displaystyle D_{x}f(x)\,}"></span>.</dd></dl> <p>Notasi Euler ini sering digunakan dalam menyelesaikan <a href="/wiki/Persamaan_diferensial" title="Persamaan diferensial">persamaan diferensial linear</a>. </p> <table class="prettytable"> <tbody><tr> <th> </th> <th align="center">Notasi Leibniz </th> <th align="center">Notasi Lagrange </th> <th align="center">Notasi Newton </th> <th align="center">Notasi Euler </th></tr> <tr align="center"> <td><b>Turunan <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> terhadap <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span></b> </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 {\frac {d}{dx}}f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6818ebd87a15d28471e6742720ed7820c79c0a95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.799ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dx}}f(x)}"></span> </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 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> </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/d0cd7d7c75340e779d82658e19d1720ce84ab127" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.144ex; height:3.009ex;" alt="{\displaystyle f'(x)}"></span> </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 {\dot {y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea068ce646833369cccf19795f23613159b5f89f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.302ex; height:2.509ex;" alt="{\displaystyle {\dot {y}}}"></span><br /> dengan <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=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2311a6a75c54b0ea085a381ba472c31d59321514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.672ex; height:2.843ex;" alt="{\displaystyle y=f(x)}"></span> </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 D_{x}f(x)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{x}f(x)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3728d37766091b7849295bcc2ac2c895eaef0b0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.901ex; height:2.843ex;" alt="{\displaystyle D_{x}f(x)\,}"></span> </td></tr></tbody></table> <div style="clear:both;"></div> <div class="mw-heading mw-heading3"><h3 id="Integral">Integral</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18844875"><div role="note" class="hatnote navigation-not-searchable">Artikel utama: <a href="/wiki/Integral" title="Integral">Integral</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Berkas: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/250px-Integral_as_region_under_curve.svg.png" decoding="async" width="250" height="233" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Integral_as_region_under_curve.svg/375px-Integral_as_region_under_curve.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Integral_as_region_under_curve.svg/500px-Integral_as_region_under_curve.svg.png 2x" data-file-width="750" data-file-height="700" /></a><figcaption>Integral dapat dianggap sebagai perhitungan luas daerah di bawah kurva <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>, antara dua titik <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/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> dan <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/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>.</figcaption></figure> <p>Integral merupakan suatu objek matematika yang dapat diinterpretasikan sebagai luas wilayah ataupun generalisasi suatu wilayah. Proses menemukan integral suatu fungsi disebut sebagai pengintegralan ataupun integrasi. Integral dibagi menjadi dua, yaitu: integral tertentu dan integral tak tentu. Notasi matematika yang digunakan untuk menyatakan integral adalah <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 \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2794e142343d3783e2163fcebe1f6bcb028f9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:2.581ex; height:5.676ex;" alt="{\displaystyle \int \,}"></span>, seperti huruf S yang memanjang (S singkatan dari <i>"Sum"</i> yang berarti penjumlahan).<sup id="cite_ref-concepts_1-5" class="reference"><a href="#cite_note-concepts-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Integral_tertentu">Integral tertentu</h4></div> <p>Diberikan suatu fungsi <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> bervariabel real <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> dan interval antara <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> pada garis real, <b>integral tertentu</b>: </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> <mspace width="thinmathspace" /> <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/2116609827d40b296a5a4670a86038166b5b7546" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.173ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(x)\,dx\,,}"></span></dd></dl> <p>secara informal didefinisikan sebagai luas daerah pada bidang-<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 xy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72eb345e496513fb8b2fa4aa8c4d89b855f9a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle xy}"></span> yang dibatasi oleh kurva grafik <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>, sumbu-<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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>, dan garis vertikal <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=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaae23950e96a955ab5b07015a168fd931d4d82b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.658ex; height:1.676ex;" alt="{\displaystyle x=a}"></span> dan <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/229dd3f0f42d375fa257bdc1389f92f7b225c415" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.426ex; height:2.176ex;" alt="{\displaystyle x=b}"></span>. </p><p>Pada notasi integral di atas: <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/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> adalah <i>batas bawah</i> dan <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/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> adalah <i>batas atas</i> yang menentukan domain pengintegralan, <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> adalah integran yang akan dievaluasi terhadap <i><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span></i> pada 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>, dan <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/845c817e348381a13f3fad5184169ce0e021c685" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.546ex; height:2.176ex;" alt="{\displaystyle dx}"></span> adalah variabel pengintegralan. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Riemann.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Riemann.gif/250px-Riemann.gif" decoding="async" width="250" height="134" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Riemann.gif/375px-Riemann.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Riemann.gif/500px-Riemann.gif 2x" data-file-width="598" data-file-height="321" /></a><figcaption>Seiring dengan semakin banyaknya subinterval dan semakin sempitnya lebar subinterval yang diambil, luas keseluruhan batangan akan semakin mendekati luas daerah di bawah kurva.</figcaption></figure> <p>Terdapat berbagai jenis pendefinisian formal integral tertentu, namun yang paling umumnya digunakan adalah definisi <a href="/wiki/Integral_Riemann" title="Integral Riemann">integral Riemann</a>. Integral Riemann didefinisikan sebagai limit dari "<a href="/wiki/Jumlah_Riemann" title="Jumlah Riemann">penjumlahan Riemann</a>". Misalkan ingin mencari luas daerah yang dibatasi oleh fungsi <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> pada interval tertutup <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>. Dalam mencari luas daerah tersebut, 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> dapat dibagi menjadi banyak subinterval yang lebarnya tidak perlu sama, dan memilih sejumlah <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 n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd0b0f32b28f51962943ee9ede4fb34198a2521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}"></span> titik <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_{1},x_{2},x_{3},\dots ,x_{n-1}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</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 fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x_{1},x_{2},x_{3},\dots ,x_{n-1}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aca2a5c7459dbb024a78a25dcaef41ed90079f25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.371ex; height:2.843ex;" alt="{\displaystyle \{x_{1},x_{2},x_{3},\dots ,x_{n-1}\}}"></span> antara <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/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> dengan <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/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> sehingga memenuhi hubungan:<sup id="cite_ref-riemann_18-0" class="reference"><a href="#cite_note-riemann-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><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 x_{1}\leq x_{2}\leq \cdots \leq x_{n-1}\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>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</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>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 x_{1}\leq x_{2}\leq \cdots \leq x_{n-1}\leq x_{n}=b.\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc5fcb407c60cb5bb72ec0046e1355e55c73ebda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:42.022ex; height:2.509ex;" alt="{\displaystyle a=x_{0}\leq x_{1}\leq x_{2}\leq \cdots \leq x_{n-1}\leq x_{n}=b.\,\!}"></span></dd></dl></dd></dl> <p>Himpunan <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 P=\{x_{0},x_{1},x_{2},\ldots ,x_{n-1},x_{n}\}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</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>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=\{x_{0},x_{1},x_{2},\ldots ,x_{n-1},x_{n}\}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e023d04cc26e5e8637c27171bb12e2c65beafa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.185ex; height:2.843ex;" alt="{\displaystyle P=\{x_{0},x_{1},x_{2},\ldots ,x_{n-1},x_{n}\}\,}"></span> tersebut dapat dikatakan sebagai <b>partisi</b> <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>, yang membagi <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> menjadi sejumlah <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> subinterval <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_{0},x_{1}],[x_{1},x_{2}],\ldots ,[x_{n-1},x_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>,</mo> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x_{0},x_{1}],[x_{1},x_{2}],\ldots ,[x_{n-1},x_{n}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6e703c312490f26ff6dd5e474f67c6606b57b6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.928ex; height:2.843ex;" alt="{\displaystyle [x_{0},x_{1}],[x_{1},x_{2}],\ldots ,[x_{n-1},x_{n}]}"></span>. Lebar subinterval pertama <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_{0},x_{1}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x_{0},x_{1}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec10faf54bec6a09b997bebef2b4417ec2ebc8b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.095ex; height:2.843ex;" alt="{\displaystyle [x_{0},x_{1}]}"></span> dinyatakan sebagai <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 x_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/679f8c97b41fcd56414b4061b706499bc77deb7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.32ex; height:2.509ex;" alt="{\displaystyle \Delta x_{1}}"></span>, demikian pula lebar subinterval ke-<i>i</i> dinyatakan sebagai <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 x_{i}=x_{i}-x_{i-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x_{i}=x_{i}-x_{i-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a96f1ce8a935fd3a57f1b189f5eb056b85581906" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.363ex; height:2.509ex;" alt="{\displaystyle \Delta x_{i}=x_{i}-x_{i-1}}"></span>. Pada tiap-tiap subinterval inilah dipilih suatu titik sembarang, dan pada subinterval ke-<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 i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> tersebut dipilih titik sembarang <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 t_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b61e3d4d909be4a19c9a554a301684232f59e5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.639ex; height:2.343ex;" alt="{\displaystyle t_{i}}"></span>. Maka pada tiap-tiap subinterval akan terdapat batangan persegi panjang yang lebarnya sebesar <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3890eb866b6258d7a304fc34c70ee3fb3a81a70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.266ex; height:2.176ex;" alt="{\displaystyle \Delta x}"></span> dan tingginya berawal dari sumbu <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> sampai menyentuh titik <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 (t_{i},f(t_{i}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <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> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (t_{i},f(t_{i}))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70cfad999694f2b22f286fd06e6c59d53bf0aba1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.21ex; height:2.843ex;" alt="{\displaystyle (t_{i},f(t_{i}))}"></span> pada kurva. Jika luas tiap-tiap batangan tersebut dihitung dengan mengalikan <i>ƒ</i>(<i>t</i><sub>i</sub>)· Δ<i>x</i><sub>i</sub> dan menjumlahkan keseluruhan luas daerah batangan tersebut, maka akan didapatkan: </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_{p}=\sum _{i=1}^{n}f(t_{i})\Delta x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <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> <mi mathvariant="normal">Δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{p}=\sum _{i=1}^{n}f(t_{i})\Delta x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bd6514e27d91f096fb3ac96fa13098c7258c121" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.117ex; height:6.843ex;" alt="{\displaystyle S_{p}=\sum _{i=1}^{n}f(t_{i})\Delta x_{i}}"></span></dd></dl> <p>Penjumlahan <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_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/753062ca2b97967794cc23e3e553f46898493d8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.484ex; height:2.843ex;" alt="{\displaystyle S_{p}}"></span> disebut sebagai <b>penjumlahan Riemann untuk <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> pada 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>.</b> Perhatikan bahwa semakin kecil subinterval partisi yang diambil, hasil penjumlahan Riemann ini akan semakin mendekati nilai luas daerah yang diinginkan. Jika limit dari norma partisi <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 \lVert P\rVert }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>P</mi> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lVert P\rVert }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/332e3f1e24199a506fc98412f99ec3fd5ff22431" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.07ex; height:2.843ex;" alt="{\displaystyle \lVert P\rVert }"></span> mendekati nol, maka didapatkan luas daerah tersebut.<sup id="cite_ref-riemann_18-1" class="reference"><a href="#cite_note-riemann-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>Secara cermat, definisi integral tertentu sebagai limit dari penjumlahan Riemann adalah:<sup id="cite_ref-riemann_18-2" class="reference"><a href="#cite_note-riemann-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r21477316"><blockquote class="templatequote"><p>Diberikan <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> sebagai fungsi yang terdefinisikan pada interval tertutup <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>. Bilangan <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 I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> dikatakan sebagai <b>integral tertentu</b> <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> di sepanjang <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> dan bahwa <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 I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> adalah limit dari penjumlahan Riemann <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 x_{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> <mi mathvariant="normal">Δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}f(t_{i})\Delta x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/398af9d6a56b8afd5254df5a4645a4f614f94e88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.534ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{n}f(t_{i})\Delta x_{i}}"></span> jikamemenuhi syarat berikut: Untuk setiap bilangan <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>, terdapat sebuah bilangan <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> yang berkorespondensi dengannya sedemikian rupa untuk setiap partisi <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 P=\{x_{0},x_{1},\ldots ,x_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=\{x_{0},x_{1},\ldots ,x_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ebbf8880b61f64c35a4760df3ddbf6ff5806b82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.697ex; height:2.843ex;" alt="{\displaystyle P=\{x_{0},x_{1},\ldots ,x_{n}\}}"></span> di sepanjang <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> dengan <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 \lVert P\rVert <\delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>P</mi> <mo fence="false" stretchy="false">‖</mo> <mo><</mo> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lVert P\rVert <\delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/067a08c58db56c5e47764bedf8ebea6da8a55773" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.217ex; height:2.843ex;" alt="{\displaystyle \lVert P\rVert <\delta }"></span> dan pilihan <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 t_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b61e3d4d909be4a19c9a554a301684232f59e5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.639ex; height:2.343ex;" alt="{\displaystyle t_{i}}"></span> apapun pada <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_{k-1},t_{i}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x_{k-1},t_{i}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/333bc6a8960983ed222489324bb2fa2fe712a22d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.486ex; height:2.843ex;" alt="{\displaystyle [x_{k-1},t_{i}]}"></span>, maka didapatkan </p><dl><dd><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|\sum _{i=1}^{n}f(t_{i})\Delta x_{i}-I\right|<\varepsilon .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <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> <mi mathvariant="normal">Δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>I</mi> </mrow> <mo>|</mo> </mrow> <mo><</mo> <mi>ε</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\sum _{i=1}^{n}f(t_{i})\Delta x_{i}-I\right|<\varepsilon .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2bbafb675155a986a42ad713c3b8eb83674a61b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.669ex; height:7.176ex;" alt="{\displaystyle \left|\sum _{i=1}^{n}f(t_{i})\Delta x_{i}-I\right|<\varepsilon .}"></span></dd></dl></dd></dl></blockquote> <p>Secara matematis dapat ditulis: </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 \lim _{\lVert P\rVert \to 0}\sum _{i=1}^{n}f(t_{i})\Delta x_{i}=I=\int _{a}^{b}f(x)\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">‖</mo> <mi>P</mi> <mo fence="false" stretchy="false">‖</mo> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <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> <mi mathvariant="normal">Δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>I</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{\lVert P\rVert \to 0}\sum _{i=1}^{n}f(t_{i})\Delta x_{i}=I=\int _{a}^{b}f(x)\,dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12f66a56cd8023457e593472d59ed7beba628be0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:36.773ex; height:7.009ex;" alt="{\displaystyle \lim _{\lVert P\rVert \to 0}\sum _{i=1}^{n}f(t_{i})\Delta x_{i}=I=\int _{a}^{b}f(x)\,dx}"></span></dd></dl> <p>Jika masing-masing partisi mempunyai sejumlah <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> subinterval yang sama, maka lebar <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 x={\tfrac {b-a}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mi>n</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x={\tfrac {b-a}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cdde88dc66432fe23a190663533b594f814de05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.054ex; height:3.509ex;" alt="{\displaystyle \Delta x={\tfrac {b-a}{n}}}"></span>, sehingga persamaan di atas dapat pula ditulis sebagai: </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 \lim _{n\to \infty }\sum _{i=1}^{n}f(t_{i})\Delta x=I=\int _{a}^{b}f(x)\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <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> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo>=</mo> <mi>I</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }\sum _{i=1}^{n}f(t_{i})\Delta x=I=\int _{a}^{b}f(x)\,dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a91f346b1e8a5da5d7909744c5bebb4bee43bad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.902ex; height:7.009ex;" alt="{\displaystyle \lim _{n\to \infty }\sum _{i=1}^{n}f(t_{i})\Delta x=I=\int _{a}^{b}f(x)\,dx}"></span></dd></dl> <p>Limit ini selalu diambil ketika norma partisi mendekati nol dan jumlah subinterval yang ada mendekati tak terhingga banyaknya.<sup id="cite_ref-riemann_18-3" class="reference"><a href="#cite_note-riemann-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <dl><dt><b>Contoh</b></dt></dl> <p>Sebagai contoh, jika integral tertentu <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}^{b}x\,dx}"> <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>b</mi> </mrow> </msubsup> <mi>x</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{b}x\,dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e17e587b3dce63e2b800c7ceff808ea9ebc6de8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.051ex; height:6.343ex;" alt="{\displaystyle \int _{0}^{b}x\,dx}"></span> dihitung untuk mencari luas daerah <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> di bawah kurva <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0abe2e7da593fb7b41d44e87a97fefdd8998b77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.009ex;" alt="{\displaystyle y=x}"></span> pada 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 [0,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba22e25e8f8604f012c599a7d4962562c4bb3f02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.488ex; height:2.843ex;" alt="{\displaystyle [0,b]}"></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="{\displaystyle b>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94436473a90bd55191a79c59474cb5456dcbec00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.258ex; height:2.176ex;" alt="{\displaystyle b>0}"></span>, maka perhitungan integral tertentu <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}^{b}x\,dx}"> <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>b</mi> </mrow> </msubsup> <mi>x</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{b}x\,dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e17e587b3dce63e2b800c7ceff808ea9ebc6de8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.051ex; height:6.343ex;" alt="{\displaystyle \int _{0}^{b}x\,dx}"></span> sebagai limit dari penjumlahan Riemannnya adalah <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 \lim _{\lVert P\rVert \to 0}\sum _{i=1}^{n}f(t_{i})\Delta x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">‖</mo> <mi>P</mi> <mo fence="false" stretchy="false">‖</mo> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <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> <mi mathvariant="normal">Δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{\lVert P\rVert \to 0}\sum _{i=1}^{n}f(t_{i})\Delta x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4801d6950d4b91f4bd1e8f67c1bbfaa0a12421f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.265ex; height:6.843ex;" alt="{\displaystyle \lim _{\lVert P\rVert \to 0}\sum _{i=1}^{n}f(t_{i})\Delta x_{i}}"></span> </p><p>Pemilihan partisi ataupun titik <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 t_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b61e3d4d909be4a19c9a554a301684232f59e5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.639ex; height:2.343ex;" alt="{\displaystyle t_{i}}"></span> secara sembarang akan menghasilkan nilai yang sama sepanjang norma partisi tersebut mendekati nol. Jika partisi <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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> yang dipilih membagi-bagi 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 [0,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba22e25e8f8604f012c599a7d4962562c4bb3f02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.488ex; height:2.843ex;" alt="{\displaystyle [0,b]}"></span> menjadi <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> subinterval yang berlebar sama <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 x={\tfrac {b-0}{n}}={\tfrac {b}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mn>0</mn> </mrow> <mi>n</mi> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>b</mi> <mi>n</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x={\tfrac {b-0}{n}}={\tfrac {b}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1de16670837e20f2a046f89a645387ccd6043dc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.927ex; height:3.509ex;" alt="{\displaystyle \Delta x={\tfrac {b-0}{n}}={\tfrac {b}{n}}}"></span> dan titik <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 t_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b61e3d4d909be4a19c9a554a301684232f59e5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.639ex; height:2.343ex;" alt="{\displaystyle t_{i}}"></span> yang dipilih adalah titik akhir kiri setiap subinterval, partisi yang didapatkan adalah <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 P=\{0,{\tfrac {b}{n}},{\tfrac {2b}{n}},{\tfrac {3b}{n}},\ldots ,{\tfrac {nb}{n}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>b</mi> <mi>n</mi> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>b</mi> </mrow> <mi>n</mi> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>b</mi> </mrow> <mi>n</mi> </mfrac> </mstyle> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>n</mi> <mi>b</mi> </mrow> <mi>n</mi> </mfrac> </mstyle> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=\{0,{\tfrac {b}{n}},{\tfrac {2b}{n}},{\tfrac {3b}{n}},\ldots ,{\tfrac {nb}{n}}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35161744a954b8039b195b87541807f5fb88b2c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.688ex; height:3.509ex;" alt="{\displaystyle P=\{0,{\tfrac {b}{n}},{\tfrac {2b}{n}},{\tfrac {3b}{n}},\ldots ,{\tfrac {nb}{n}}\}}"></span> dan <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 t_{i}={\tfrac {ib}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>i</mi> <mi>b</mi> </mrow> <mi>n</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{i}={\tfrac {ib}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9af3000525f43d18c607090dc48d831c9a8fed5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.847ex; height:3.509ex;" alt="{\displaystyle t_{i}={\tfrac {ib}{n}}}"></span>, sehingga: </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 _{0}^{b}f(x)\,dx&=\lim _{n\to \infty }\sum _{i=1}^{n}f(t_{i})\Delta x\\&=\lim _{n\to \infty }\sum _{i=1}^{n}{\frac {ib}{n}}.{\frac {b}{n}}\\&=\lim _{n\to \infty }\sum _{i=1}^{n}{\frac {ib^{2}}{n^{2}}}\\&=\lim _{n\to \infty }{\frac {b^{2}}{n^{2}}}\sum _{i=1}^{n}i\\&=\lim _{n\to \infty }{\frac {b^{2}}{n^{2}}}.{\frac {n(n+1)}{2}}\\&=\lim _{n\to \infty }{\frac {b^{2}}{2}}(1+{\frac {1}{n}})\\\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"> <mn>0</mn> </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> <mtd> <mi></mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <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> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mi>b</mi> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>.</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>n</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <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>i</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int _{0}^{b}f(x)\,dx&=\lim _{n\to \infty }\sum _{i=1}^{n}f(t_{i})\Delta x\\&=\lim _{n\to \infty }\sum _{i=1}^{n}{\frac {ib}{n}}.{\frac {b}{n}}\\&=\lim _{n\to \infty }\sum _{i=1}^{n}{\frac {ib^{2}}{n^{2}}}\\&=\lim _{n\to \infty }{\frac {b^{2}}{n^{2}}}\sum _{i=1}^{n}i\\&=\lim _{n\to \infty }{\frac {b^{2}}{n^{2}}}.{\frac {n(n+1)}{2}}\\&=\lim _{n\to \infty }{\frac {b^{2}}{2}}(1+{\frac {1}{n}})\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f438e767305198b8b110df47edfb805d4059fb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -19.505ex; width:33.405ex; height:40.176ex;" alt="{\displaystyle {\begin{aligned}\int _{0}^{b}f(x)\,dx&=\lim _{n\to \infty }\sum _{i=1}^{n}f(t_{i})\Delta x\\&=\lim _{n\to \infty }\sum _{i=1}^{n}{\frac {ib}{n}}.{\frac {b}{n}}\\&=\lim _{n\to \infty }\sum _{i=1}^{n}{\frac {ib^{2}}{n^{2}}}\\&=\lim _{n\to \infty }{\frac {b^{2}}{n^{2}}}\sum _{i=1}^{n}i\\&=\lim _{n\to \infty }{\frac {b^{2}}{n^{2}}}.{\frac {n(n+1)}{2}}\\&=\lim _{n\to \infty }{\frac {b^{2}}{2}}(1+{\frac {1}{n}})\\\end{aligned}}}"></span></dd></dl> <p>Seiring dengan <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> mendekati tak terhingga dan norma partisi <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 \lVert P\rVert }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>P</mi> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lVert P\rVert }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/332e3f1e24199a506fc98412f99ec3fd5ff22431" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.07ex; height:2.843ex;" alt="{\displaystyle \lVert P\rVert }"></span> mendekati 0, maka didapatkan: </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}^{b}f(x)\,dx=A={\frac {b^{2}}{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>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>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{b}f(x)\,dx=A={\frac {b^{2}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7733d891ce577314c91f1affbd443df2b3a65b0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.967ex; height:6.343ex;" alt="{\displaystyle \int _{0}^{b}f(x)\,dx=A={\frac {b^{2}}{2}}}"></span></dd></dl> <p>Dalam praktiknya, penerapan definisi integral tertentu dalam mencari nilai integral tertentu tersebut jarang sekali digunakan karena tidak praktis. <a class="mw-selflink-fragment" href="#Teorema_dasar">Teorema dasar kalkulus</a> (<a class="mw-selflink-fragment" href="#Teorema_dasar">lihat bagian bawah)</a> memberikan cara yang lebih praktis dalam mencari nilai integral tertentu.<sup id="cite_ref-concepts_1-6" class="reference"><a href="#cite_note-concepts-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Integral_tak_tentu">Integral tak tentu</h4></div> <p>Manakala integral tertentu adalah sebuah bilangan yang besarnya ditentukan dengan mengambil limit penjumlahan Riemann, yang diasosiasikan dengan partisi interval tertutup yang norma partisinya mendekati nol, <a href="/wiki/Teorema_dasar_kalkulus" title="Teorema dasar kalkulus">teorema dasar kalkulus</a> (<a class="mw-selflink-fragment" href="#teorema_dasar_kalkulus">lihat bagian bawah</a>) menyatakan bahwa integral tertentu sebuah fungsi kontinu dapat dihitung dengan mudah jika antiturunan/antiderivatif fungsi tersebut dapat dicari melalui teorema berikut.<sup id="cite_ref-concepts_1-7" class="reference"><a href="#cite_note-concepts-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r21477316"><blockquote class="templatequote"><p>Jika </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)={\frac {d}{dx}}F(x)=f(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>F</mi> <mo>′</mo> </msup> <mspace width="negativethinmathspace" /> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>F</mi> <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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F'\!(x)={\frac {d}{dx}}F(x)=f(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aded66280d6659a68c67aec0b2cbd371bdc3f21c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:24.774ex; height:5.509ex;" alt="{\displaystyle F'\!(x)={\frac {d}{dx}}F(x)=f(x).}"></span></dd></dl> <p>maka keseluruhan himpunan <b>antiturunan</b>/<b>antiderivatif</b> sebuah fungsi <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> adalah <b>integral tak tentu</b> ataupun <b>primitif</b> dari <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> terhadap <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> dan dituliskan secara matematis sebagai: </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=F(x)+C}"> <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> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int f(x)dx=F(x)+C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c47bff206ee7448cc31e531fc8903913ba29b75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.129ex; height:5.676ex;" alt="{\displaystyle \int f(x)dx=F(x)+C}"></span></dd></dl></blockquote> <p>Bentuk <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)+C}"> <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> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)+C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84bfd7719983cec6643eb997d3aa006ad1c3bf26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.486ex; height:2.843ex;" alt="{\displaystyle F(x)+C}"></span> adalah <b>antiderivatif umum</b> <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> dan <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> adalah konstanta sembarang. </p><p>Misalkan terdapat sebuah fungsi <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)=x^{2}}"> <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>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84ddac4ae10b1aa4a11741c79771a583419fb1fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.9ex; height:3.176ex;" alt="{\displaystyle f(x)=x^{2}}"></span>, maka integral tak tentu ataupun antiturunan dari fungsi tersebut adalah: </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 x^{2}dx={\frac {1}{3}}x^{3}+C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int x^{2}dx={\frac {1}{3}}x^{3}+C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c73c4746231ba8b0bf2e072c5b9908fd532b0561" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.598ex; height:5.676ex;" alt="{\displaystyle \int x^{2}dx={\frac {1}{3}}x^{3}+C}"></span>.</dd></dl> <p>Perhatikan bahwa integral tertentu berbeda dengan integral tak tentu. Integral tertentu dalam bentuk <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> adalah sebuah bilangan, manakala integral tak tentu:<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> </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/a913ba9ae058ed77b5335bce88eb2fcba92d3351" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.931ex; height:5.676ex;" alt="{\displaystyle \int f(x)\,dx}"></span> adalah sebuah fungsi yang memiliki tambahan konstanta sembarang <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>. </p><p><br /> </p> <div class="mw-heading mw-heading3"><h3 id="Teorema_dasar">Teorema dasar</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18844875"><div role="note" class="hatnote navigation-not-searchable">Artikel utama: <a href="/wiki/Teorema_dasar_kalkulus" title="Teorema dasar kalkulus">Teorema dasar kalkulus</a></div> <p>Teorema dasar kalkulus menyatakan bahwa turunan dan integral adalah dua operasi yang saling berlawanan. Lebih tepatnya, teorema ini menghubungkan nilai dari anti derivatif dengan integral tertentu. Karena lebih mudah menghitung sebuah anti derivatif daripada menerapkan definisi integral tertentu, teorema dasar kalkulus memberikan cara yang praktis dalam menghitung integral tertentu.<sup id="cite_ref-concepts_1-8" class="reference"><a href="#cite_note-concepts-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Teorema dasar kalkulus menyatakan: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r21477316"><blockquote class="templatequote"><p>Jika sebuah fungsi <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> adalah <a href="/wiki/Fungsi_kontinu" title="Fungsi kontinu">kontinu</a> pada 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> dan jika <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> adalah fungsi yang mana turunannya adalah <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> pada 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/d7e5710198f33b00695903460983021e75860e2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.071ex; height:2.843ex;" alt="{\displaystyle (a,b)}"></span>, maka </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>Lebih lanjut, untuk setiap <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> di 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/d7e5710198f33b00695903460983021e75860e2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.071ex; height:2.843ex;" alt="{\displaystyle (a,b)}"></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)={\frac {d}{dx}}\int _{a}^{x}f(t)\,dt=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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <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> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F'(x)={\frac {d}{dx}}\int _{a}^{x}f(t)\,dt=f(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f47c5da58709512e55851826db523c9475eff4ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.062ex; height:5.843ex;" alt="{\displaystyle F'(x)={\frac {d}{dx}}\int _{a}^{x}f(t)\,dt=f(x).}"></span></dd></dl></blockquote> <p>Sebagai contoh, jika ingin menghitung nilai 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 _{a}^{b}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>x</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}x\,dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c66be06bd12ba63995c77ffbba19716a806e2018" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.051ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}x\,dx}"></span>, daripada menggunakan definisi integral tertentu sebagai limit dari penjumlahan Riemann (<a class="mw-selflink-fragment" href="#integral_tertentu">lihat bagian atas</a>), maka teorema dasar kalkulus dapat digunakan dalam menghitung nilai integral tersebut. </p><p>Antiderivatif dari fungsi <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)=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> <mi>x</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=x\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da255cc9a1dc2e0eedeee4bdfb6025f7c308952d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.233ex; height:2.843ex;" alt="{\displaystyle f(x)=x\,}"></span> adalah <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)={\tfrac {1}{2}}x^{2}+C}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)={\tfrac {1}{2}}x^{2}+C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/009e9058c22d8ce5e50b204efb4c8109608f9fab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.627ex; height:3.509ex;" alt="{\displaystyle F(x)={\tfrac {1}{2}}x^{2}+C}"></span>. Oleh sebab itu, menurut dengan teorema dasar kalkulus, nilai dari integral tertentu <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}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>x</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}x\,dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c66be06bd12ba63995c77ffbba19716a806e2018" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.051ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}x\,dx}"></span> adalah: </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}^{b}x\,dx&=F(b)-F(a)\\&={\frac {1}{2}}b^{2}-{\frac {1}{2}}a^{2}\\\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>b</mi> </mrow> </msubsup> <mi>x</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> <mtd> <mi></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> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int _{a}^{b}x\,dx&=F(b)-F(a)\\&={\frac {1}{2}}b^{2}-{\frac {1}{2}}a^{2}\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5339a669494ea56800717e0254f13291a934201" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:24.069ex; height:11.843ex;" alt="{\displaystyle {\begin{aligned}\int _{a}^{b}x\,dx&=F(b)-F(a)\\&={\frac {1}{2}}b^{2}-{\frac {1}{2}}a^{2}\\\end{aligned}}}"></span></dd></dl> <p>Jika ingin mencari luas daerah <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> terhadap kurva <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0abe2e7da593fb7b41d44e87a97fefdd8998b77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.009ex;" alt="{\displaystyle y=x}"></span> pada 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 [0,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba22e25e8f8604f012c599a7d4962562c4bb3f02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.488ex; height:2.843ex;" alt="{\displaystyle [0,b]}"></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="{\displaystyle b>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94436473a90bd55191a79c59474cb5456dcbec00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.258ex; height:2.176ex;" alt="{\displaystyle b>0}"></span>, maka didapatkan: </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}^{b}x\,dx={\frac {b^{2}}{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>b</mi> </mrow> </msubsup> <mi>x</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{b}x\,dx={\frac {b^{2}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6e3b28d1c85b06b58f0afeb641fd4d0ff125bf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.038ex; height:6.343ex;" alt="{\displaystyle \int _{0}^{b}x\,dx={\frac {b^{2}}{2}}}"></span></dd></dl> <p>Perhatikan bahwa hasil yang didapatkan dengan menggunakan teorema dasar kalkulus ini adalah sama dengan hasil yang didapatkan dengan menerapkan definisi integral tertentu (<a class="mw-selflink-fragment" href="#integral_tertentu">lihat bagian atas</a>). Oleh karena lebih praktis, teorema dasar kalkulus sering digunakan untuk mencari nilai integral tertentu.<sup id="cite_ref-concepts_1-9" class="reference"><a href="#cite_note-concepts-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Aplikasi">Aplikasi</h2></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Berkas:NautilusCutawayLogarithmicSpiral.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/NautilusCutawayLogarithmicSpiral.jpg/220px-NautilusCutawayLogarithmicSpiral.jpg" decoding="async" width="220" height="166" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/NautilusCutawayLogarithmicSpiral.jpg/330px-NautilusCutawayLogarithmicSpiral.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/08/NautilusCutawayLogarithmicSpiral.jpg/440px-NautilusCutawayLogarithmicSpiral.jpg 2x" data-file-width="2240" data-file-height="1693" /></a><figcaption>Pola spiral logaritma cangkang Nautilus adalah contoh klasik untuk menggambarkan perkembangan dan perubahan yang berkaitan dengan kalkulus.</figcaption></figure> <p>Kalkulus digunakan di setiap cabang sains fisik, sains komputer, <a href="/wiki/Statistika" title="Statistika">statistika</a>, <a href="/wiki/Teknik" class="mw-redirect" title="Teknik">teknik</a>, <a href="/wiki/Ekonomi" title="Ekonomi">ekonomi</a>, <a href="/wiki/Bisnis" title="Bisnis">bisnis</a>, <a href="/wiki/Kedokteran" title="Kedokteran">kedokteran</a>, <a href="/wiki/Demografi" title="Demografi">kependudukan</a>, dan di bidang-bidang lainnya. Setiap konsep di <a href="/wiki/Mekanika_klasik" title="Mekanika klasik">mekanika klasik</a> saling berhubungan melalui kalkulus. <a href="/wiki/Massa" title="Massa">Massa</a> dari sebuah benda dengan <a href="/wiki/Massa_jenis" title="Massa jenis">massa jenis</a> yang tidak diketahui, <a href="/wiki/Momen_inersia" title="Momen inersia">momen inersia</a> dari suatu objek, dan total energi dari sebuah objek dapat ditentukan dengan menggunakan kalkulus.<sup id="cite_ref-concepts_1-10" class="reference"><a href="#cite_note-concepts-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Dalam subdisiplin <a href="/wiki/Listrik" title="Listrik">listrik</a> dan <a href="/wiki/Magnetisme" title="Magnetisme">magnetisme</a>, kalkulus dapat digunakan untuk mencari total aliran (fluks) dari sebuah <a href="/wiki/Medan_elektromagnetik" title="Medan elektromagnetik">medan elektromagnetik</a>. Contoh historis lainnya adalah penggunaan kalkulus di <a href="/wiki/Hukum_gerak_Newton" title="Hukum gerak Newton">hukum gerak Newton</a>, dinyatakan sebagai <i>laju perubahan</i> yang merujuk pada turunan: <b>Laju perubahan</b> <i>momentum dari sebuah benda adalah sama dengan resultan gaya yang bekerja pada benda tersebut dengan arah yang sama.</i><sup id="cite_ref-concepts_1-11" class="reference"><a href="#cite_note-concepts-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Bahkan rumus umum dari hukum kedua Newton: Gaya = Massa × Percepatan, menggunakan perumusan kalkulus diferensial karena percepatan bisa dinyatakan sebagai turunan dari kecepatan. <a href="/wiki/Persamaan_Maxwell" title="Persamaan Maxwell">Teori elektromagnetik Maxwell</a> dan teori relativitas <a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a> juga dirumuskan menggunakan kalkulus diferensial.<sup id="cite_ref-concepts_1-12" class="reference"><a href="#cite_note-concepts-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Lihat_pula">Lihat pula</h2></div> <div class="noprint metadata navbox" role="navigation" aria-label="Portals" style="font-weight:bold;padding:0.4em 0em"><ul style="margin:0.1em 0 0"><li style="display:inline"><span style="display:inline-block;white-space:nowrap"><span style="margin:0 0.5em"><span typeof="mw:File"><a href="/wiki/Berkas:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/21px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="21" height="21" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/32px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, 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.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-columns references-column-width reflist-columns-2" style=""> <ol class="references"> <li id="cite_note-concepts-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-concepts_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-concepts_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-concepts_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-concepts_1-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-concepts_1-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-concepts_1-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-concepts_1-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-concepts_1-7"><sup><i><b>h</b></i></sup></a> <a href="#cite_ref-concepts_1-8"><sup><i><b>i</b></i></sup></a> <a href="#cite_ref-concepts_1-9"><sup><i><b>j</b></i></sup></a> <a href="#cite_ref-concepts_1-10"><sup><i><b>k</b></i></sup></a> <a href="#cite_ref-concepts_1-11"><sup><i><b>l</b></i></sup></a> <a href="#cite_ref-concepts_1-12"><sup><i><b>m</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFLatorreKenellyReedBiggers2007" class="citation">Latorre, Donald R.; Kenelly, John W.; Reed, Iris B.; Biggers, Sherry (2007), <a rel="nofollow" class="external text" href="http://books.google.com/books?id=bQhX-3k0LS8C"><i>Calculus Concepts: An Applied Approach to the Mathematics of Change</i></a>, Cengage Learning, hlm. 2, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/0-618-78981-2" title="Istimewa:Sumber buku/0-618-78981-2">0-618-78981-2</a>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230327123024/https://books.google.com/books?id=bQhX-3k0LS8C&hl=en">diarsipkan</a> dari versi asli tanggal 2023-03-27<span class="reference-accessdate">, diakses tanggal <span class="nowrap">2013-11-08</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus+Concepts%3A+An+Applied+Approach+to+the+Mathematics+of+Change&rft.pages=2&rft.pub=Cengage+Learning&rft.date=2007&rft.isbn=0-618-78981-2&rft.aulast=Latorre&rft.aufirst=Donald+R.&rft.au=Kenelly%2C+John+W.&rft.au=Reed%2C+Iris+B.&rft.au=Biggers%2C+Sherry&rft_id=http%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DbQhX-3k0LS8C&rfr_id=info%3Asid%2Fid.wikipedia.org%3AKalkulus" class="Z3988"><span style="display:none;"> </span></span>, <a rel="nofollow" class="external text" href="http://books.google.com/books?id=bQhX-3k0LS8C&pg=PA2">Chapter 1, p 2</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230327123025/https://books.google.com/books?id=bQhX-3k0LS8C&pg=PA2&hl=en">Diarsipkan</a> 2023-03-27 di <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</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">Morris Kline, <i>Mathematical thought from ancient to modern times</i>, Vol. I</span> </li> <li id="cite_note-Aslaksen-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Aslaksen_3-0">^</a></b></span> <span class="reference-text">Helmer Aslaksen. <a rel="nofollow" class="external text" href="http://www.math.nus.edu.sg/aslaksen/teaching/calculus.html">Why Calculus?</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20101014164501/http://www.math.nus.edu.sg/aslaksen/teaching/calculus.html">Diarsipkan</a> 2010-10-14 di <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. <a href="/wiki/Universitas_Nasional_Singapura" title="Universitas Nasional Singapura">National University of Singapore</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">Archimedes, <i>Method</i>, in <i>The Works of Archimedes</i> <a href="/wiki/Istimewa:Sumber_buku/9780521661607" class="internal mw-magiclink-isbn">ISBN 978-0-521-66160-7</a></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"><cite class="citation web"><a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/Biographies/Aryabhata_I.html">"Aryabhata the Elder"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150711055702/http://www-history.mcs.st-andrews.ac.uk/Biographies/Aryabhata_I.html">Diarsipkan</a> dari versi asli tanggal 2015-07-11<span class="reference-accessdate">. Diakses tanggal <span class="nowrap">2007-08-09</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Aryabhata+the+Elder&rft_id=http%3A%2F%2Fwww-history.mcs.st-andrews.ac.uk%2FBiographies%2FAryabhata_I.html&rfr_id=info%3Asid%2Fid.wikipedia.org%3AKalkulus" class="Z3988"><span style="display:none;"> </span></span></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">Ian G. Pearce. <a rel="nofollow" class="external text" href="http://turnbull.mcs.st-and.ac.uk/~history/Projects/Pearce/Chapters/Ch8_5.html">Bhaskaracharya II.</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160901092504/http://turnbull.mcs.st-and.ac.uk/~history/Projects/Pearce/Chapters/Ch8_5.html">Diarsipkan</a> 2016-09-01 di <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</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">Victor J. Katz (1995). "Ideas of Calculus in Islam and India", <i>Mathematics Magazine</i> <b>68</b> (3), hlm. 163-174.</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">J. L. Berggren (1990). "Innovation and Tradition in Sharaf al-Din al-Tusi's Muadalat", <i>Journal of the American Oriental Society</i> <b>110</b> (2), hlm. 304-309.</span> </li> <li id="cite_note-madhava-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-madhava_9-0">^</a></b></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20060514012903/http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Madhava.html">"Madhava"</a>. <i>Biography of Madhava</i>. School of Mathematics and Statistics University of St Andrews, Scotland. Diarsipkan dari <a rel="nofollow" class="external text" href="http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Madhava.html">versi asli</a> tanggal 2006-05-14<span class="reference-accessdate">. Diakses tanggal <span class="nowrap">2006-09-13</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Biography+of+Madhava&rft.atitle=Madhava&rft_id=http%3A%2F%2Fwww-gap.dcs.st-and.ac.uk%2F~history%2FBiographies%2FMadhava.html&rfr_id=info%3Asid%2Fid.wikipedia.org%3AKalkulus" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-scotlnd-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-scotlnd_10-0">^</a></b></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/HistTopics/Indian_mathematics.html">"An overview of Indian mathematics"</a>. <i>Indian Maths</i>. School of Mathematics and Statistics University of St Andrews, Scotland. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060703002618/http://www-history.mcs.st-andrews.ac.uk/HistTopics/Indian_mathematics.html">Diarsipkan</a> dari versi asli tanggal 2006-07-03<span class="reference-accessdate">. Diakses tanggal <span class="nowrap">2006-07-07</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Indian+Maths&rft.atitle=An+overview+of+Indian+mathematics&rft_id=http%3A%2F%2Fwww-history.mcs.st-andrews.ac.uk%2FHistTopics%2FIndian_mathematics.html&rfr_id=info%3Asid%2Fid.wikipedia.org%3AKalkulus" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-pdffile3-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-pdffile3_11-0">^</a></b></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20060821195309/http://www.kerala.gov.in/keralcallsep04/p22-24.pdf">"Science and technology in free India"</a> <span style="font-size:85%;">(PDF)</span>. <i>Government of Kerala — Kerala Call, September 2004</i>. Prof.C.G.Ramachandran Nair. Diarsipkan dari <a rel="nofollow" class="external text" href="http://www.kerala.gov.in/keralcallsep04/p22-24.pdf">versi asli</a> <span style="font-size:85%;">(PDF)</span> tanggal 2006-08-21<span class="reference-accessdate">. Diakses tanggal <span class="nowrap">2006-07-09</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Government+of+Kerala+%E2%80%94+Kerala+Call%2C+September+2004&rft.atitle=Science+and+technology+in+free+India&rft_id=http%3A%2F%2Fwww.kerala.gov.in%2Fkeralcallsep04%2Fp22-24.pdf&rfr_id=info%3Asid%2Fid.wikipedia.org%3AKalkulus" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-charles-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-charles_12-0">^</a></b></span> <span class="reference-text"><cite class="citation book">Charles Whish (1835). <i>Transactions of the Royal Asiatic Society of Great Britain and Ireland</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Transactions+of+the+Royal+Asiatic+Society+of+Great+Britain+and+Ireland&rft.date=1835&rft.au=Charles+Whish&rfr_id=info%3Asid%2Fid.wikipedia.org%3AKalkulus" class="Z3988"><span style="display:none;"> </span></span></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"><cite class="citation book"><a rel="nofollow" class="external text" href="http://www.archive.org/details/geometricallectu00barruoft"><i>The geometrical lectures of Isaac Barrow, translated, with notes and proofs, and a discussion on the advance made therein on the work of his predecessors in the infinitesimal calculus</i></a>. Chicago: Open Court. 1916.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+geometrical+lectures+of+Isaac+Barrow%2C+translated%2C+with+notes+and+proofs%2C+and+a+discussion+on+the+advance+made+therein+on+the+work+of+his+predecessors+in+the+infinitesimal+calculus&rft.place=Chicago&rft.pub=Open+Court&rft.date=1916&rft_id=http%3A%2F%2Fwww.archive.org%2Fdetails%2Fgeometricallectu00barruoft&rfr_id=info%3Asid%2Fid.wikipedia.org%3AKalkulus" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Simmons-14"><span class="mw-cite-backlink">^ <a href="#cite_ref-Simmons_14-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Simmons_14-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Simmons_14-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Simmons_14-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Simmons_14-4"><sup><i><b>e</b></i></sup></a></span> <span class="reference-text"><cite class="citation book">Simmons, George F. (2007). <a rel="nofollow" class="external text" href="https://archive.org/details/calculusgemsbrie0000simm"><i>Calculus Gems: Brief Lives and Memorable Mathematics</i></a>. Mathematical Association of America. hlm. <a rel="nofollow" class="external text" href="https://archive.org/details/calculusgemsbrie0000simm/page/98">98</a>. <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/0-88385-561-5" title="Istimewa:Sumber buku/0-88385-561-5">0-88385-561-5</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus+Gems%3A+Brief+Lives+and+Memorable+Mathematics&rft.pages=98&rft.pub=Mathematical+Association+of+America&rft.date=2007&rft.isbn=0-88385-561-5&rft.aulast=Simmons&rft.aufirst=George+F.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcalculusgemsbrie0000simm&rfr_id=info%3Asid%2Fid.wikipedia.org%3AKalkulus" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-leibniz-15"><span class="mw-cite-backlink">^ <a href="#cite_ref-leibniz_15-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-leibniz_15-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-leibniz_15-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">Leibniz, Gottfried Wilhelm. The Early Mathematical Manuscripts of Leibniz. Cosimo, Inc., 2008. hlm. 228. <a rel="nofollow" class="external text" href="http://books.google.com/books?hl=en&lr=&id=7d8_4WPc9SMC&oi=fnd&pg=PA3&dq=Gottfried+Wilhelm+Leibniz+accused+of+plagiarism+by+Newton&ots=09h9BdTlbE&sig=hu5tNKpBJxHcpj8U3kR_T2bZqrY#v=onepage&q=plagairism&f=false">Online Copy</a></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"><cite class="citation web">Unlu, Elif (1995). <a rel="nofollow" class="external text" href="http://www.agnesscott.edu/lriddle/women/agnesi.htm">"Maria Gaetana Agnesi"</a>. Agnes Scott College. <a rel="nofollow" class="external text" href="https://web.archive.org/web/19981203075738/http://www.agnesscott.edu/lriddle/women/agnesi.htm">Diarsipkan</a> dari versi asli tanggal 1998-12-03<span class="reference-accessdate">. Diakses tanggal <span class="nowrap">2013-11-08</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Maria+Gaetana+Agnesi&rft.pub=Agnes+Scott+College&rft.date=1995&rft.aulast=Unlu&rft.aufirst=Elif&rft_id=http%3A%2F%2Fwww.agnesscott.edu%2Flriddle%2Fwomen%2Fagnesi.htm&rfr_id=info%3Asid%2Fid.wikipedia.org%3AKalkulus" class="Z3988"><span style="display:none;"> </span></span> <span style="display:none;font-size:100%" class="error citation-comment">Parameter <code style="color:inherit; border:inherit; padding:inherit;">|month=</code> yang tidak diketahui akan diabaikan (<a href="/wiki/Bantuan:Galat_CS1#parameter_ignored" title="Bantuan:Galat CS1">bantuan</a>)</span></span> </li> <li id="cite_note-Larson-17"><span class="mw-cite-backlink">^ <a href="#cite_ref-Larson_17-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Larson_17-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite class="citation book"><a href="/w/index.php?title=Ron_Larson_(mathematician)&action=edit&redlink=1" class="new" title="Ron Larson (mathematician) (halaman belum tersedia)">Larson, Ron</a>; Edwards, Bruce H. (2010). <i>Calculus of a single variable</i> (edisi ke-Ninth). <a href="/w/index.php?title=Brooks/Cole&action=edit&redlink=1" class="new" title="Brooks/Cole (halaman belum tersedia)">Brooks/Cole</a>, <a href="/w/index.php?title=Cengage_Learning&action=edit&redlink=1" class="new" title="Cengage Learning (halaman belum tersedia)">Cengage Learning</a>. <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0-547-20998-2" title="Istimewa:Sumber buku/978-0-547-20998-2">978-0-547-20998-2</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus+of+a+single+variable&rft.edition=Ninth&rft.pub=Brooks%2FCole%2C+Cengage+Learning&rft.date=2010&rft.isbn=978-0-547-20998-2&rft.aulast=Larson&rft.aufirst=Ron&rft.au=Edwards%2C+Bruce+H.&rfr_id=info%3Asid%2Fid.wikipedia.org%3AKalkulus" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-riemann-18"><span class="mw-cite-backlink">^ <a href="#cite_ref-riemann_18-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-riemann_18-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-riemann_18-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-riemann_18-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text">Bernard Riemann. "Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe" (On the representability of a function by a trigonometric series; i.e., when can a function be represented by a trigonometric series). Makalah ini diserahkan kepada Universitas Göttingen pada tahun 1854 sebagai <i>Habilitationsschrift</i> Riemann (kualifikasi untuk menjadi instruktur). Diterbitkan pada tahun 1868 dalam <i>Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen</i> (Proceedings of the Royal Philosophical Society at Göttingen), vol. 13, hlm. 87-132. (dapat dibaca <a rel="nofollow" class="external text" href="http://books.google.com/books?id=PDVFAAAAcAAJ&pg=RA1-PA87">di sini</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230327123004/https://books.google.com/books?id=PDVFAAAAcAAJ&pg=RA1-PA87&hl=en">Diarsipkan</a> 2023-03-27 di <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>..) Definisi integral Riemann, lihat bagian 4, "Über der Begriff eines bestimmten Integrals und den Umfang seiner Gültigkeit" (On the concept of a definite integral and the extent of its validity), hlm. 101-103.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Daftar_pustaka">Daftar pustaka</h2></div> <ul><li>Donald A. McQuarrie (2003). <i>Mathematical Methods for Scientists and Engineers</i>, University Science Books. <a href="/wiki/Istimewa:Sumber_buku/9781891389245" class="internal mw-magiclink-isbn">ISBN 978-1-891389-24-5</a></li> <li>James Stewart (2002). <i>Calculus: Early Transcendentals</i>, 5th ed., Brooks Cole. <a href="/wiki/Istimewa:Sumber_buku/9780534393212" class="internal mw-magiclink-isbn">ISBN 978-0-534-39321-2</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Sumber_lain">Sumber lain</h2></div> <div class="mw-heading mw-heading3"><h3 id="Bacaan_lebih_lanjut">Bacaan lebih lanjut</h3></div> <ul><li>Robert A. Adams. (1999) <a href="/wiki/Istimewa:Sumber_buku/9780201396072" class="internal mw-magiclink-isbn">ISBN 978-0-201-39607-2</a> <i>Calculus: A complete course</i>.</li> <li>Albers, Donald J.; Richard D. Anderson and Don O. Loftsgaarden, ed. (1986) <i>Undergraduate Programs in the Mathematics and Computer Sciences: The 1985-1986 Survei</i>, Mathematical Association of America No. 7,</li> <li>John L. Bell: <i>A Primer of Infinitesimal Analysis</i>, Cambridge University Press, 1998. <a href="/wiki/Istimewa:Sumber_buku/9780521624015" class="internal mw-magiclink-isbn">ISBN 978-0-521-62401-5</a>.</li> <li>Florian Cajori, "The History of Notations of the Calculus." <i>Annals of Mathematics</i>, 2nd Ser., Vol. 25, No. 1 (Sep., 1923), hlm. 1-46.</li> <li>Leonid P. Lebedev and Michael J. Cloud: "Approximating Perfection: a Mathematician's Journey into the World of Mechanics, Ch. 1: The Tools of Calculus", Princeton Univ. Press, 2004</li> <li>Cliff Pickover. (2003) <a href="/wiki/Istimewa:Sumber_buku/9780471269878" class="internal mw-magiclink-isbn">ISBN 978-0-471-26987-8</a> <i>Calculus and Pizza: A Math Cookbook for the Hungry Mind</i>.</li> <li>Michael Spivak. (Sept 1994) <a href="/wiki/Istimewa:Sumber_buku/9780914098898" class="internal mw-magiclink-isbn">ISBN 978-0-914098-89-8</a><i> Calculus</i>. Publish or Perish publishing.</li> <li>Silvanus P. Thompson dan Martin Gardner. (1998) <a href="/wiki/Istimewa:Sumber_buku/9780312185480" class="internal mw-magiclink-isbn">ISBN 978-0-312-18548-0</a> <i>Calculus Made Easy</i>.</li> <li>Mathematical Association of America. (1988) <i>Calculus for a New Century; A Pump, Not a Filter</i>, The Association, Stony Brook, NY. ED 300 252.</li> <li>Thomas/Finney. (1996) <a href="/wiki/Istimewa:Sumber_buku/9780201531749" class="internal mw-magiclink-isbn">ISBN 978-0-201-53174-9</a> <i>Calculus and Analytic geometry 9th</i>, Addison Wesley.</li> <li>Weisstein, Eric W. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/SecondFundamentalTheoremofCalculus.html">"Second Fundamental Theorem of Calculus."</a> dari MathWorld—A Wolfram Web Resource.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Pustaka_daring">Pustaka daring</h3></div> <ul><li>Crowell, B., (2003). "<i>Calculus</i>" Light and Matter, Fullerton. Retrieved 6th May 2007 from <a rel="nofollow" class="external text" href="http://www.lightandmatter.com/calc/calc.pdf">http://www.lightandmatter.com/calc/calc.pdf</a></li> <li>Garrett, P., (2006). "<i>Notes on first year calculus</i>" University of Minnesota. Retrieved 6th May 2007 from <a rel="nofollow" class="external text" href="http://www.math.umn.edu/~garrett/calculus/first_year/notes.pdf">http://www.math.umn.edu/~garrett/calculus/first_year/notes.pdf</a></li> <li>Faraz, H., (2006). "<i>Understanding Calculus</i>" Retrieved Retrieved 6th May 2007 from Understanding Calculus, URL <a rel="nofollow" class="external text" href="http://www.understandingcalculus.com/">http://www.understandingcalculus.com/</a> (HTML only)</li> <li>Keisler, H. J., (2000). "<i>Elementary Calculus: An Approach Using Infinitesimals</i>" Retrieved 6th May 2007 from <a rel="nofollow" class="external text" href="http://www.math.wisc.edu/~keisler/keislercalc1.pdf">http://www.math.wisc.edu/~keisler/keislercalc1.pdf</a></li> <li>Mauch, S. (2004). "<i>Sean's Applied Math Book</i>" California Institute of Technology. Retrieved 6th May 2007 from <a rel="nofollow" class="external text" href="http://www.cacr.caltech.edu/~sean/applied_math.pdf">http://www.cacr.caltech.edu/~sean/applied_math.pdf</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070614183657/http://www.cacr.caltech.edu/~sean/applied_math.pdf">Diarsipkan</a> 2007-06-14 di <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</li> <li>Sloughter, Dan., (2000) "<i>Difference Equations to Differential Equations: An introduction to calculus</i>". Retrieved 6th May 2007 from <a rel="nofollow" class="external text" href="http://math.furman.edu/~dcs/book/">http://math.furman.edu/~dcs/book/</a></li> <li>Stroyan, K.D., (2004). "<i>A brief introduction to infinitesimal calculus</i>" University of Iowa. Retrieved 6th May 2007 from <a rel="nofollow" class="external text" href="http://www.math.uiowa.edu/~stroyan/InfsmlCalculus/InfsmlCalc.htm">http://www.math.uiowa.edu/~stroyan/InfsmlCalculus/InfsmlCalc.htm</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20050911104158/http://www.math.uiowa.edu/~stroyan/InfsmlCalculus/InfsmlCalc.htm">Diarsipkan</a> 2005-09-11 di <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. (HTML only)</li> <li>Strang, G. (1991) "<i>Calculus</i>" Massachusetts Institute of Technology. Retrieved 6th May 2007 from <a rel="nofollow" class="external text" href="http://ocw.mit.edu/ans7870/resources/Strang/strangtext.htm">http://ocw.mit.edu/ans7870/resources/Strang/strangtext.htm</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Halaman_web">Halaman web</h3></div> <ul><li><a rel="nofollow" class="external text" href="http://www.calculus.org">Calculus.org: The Calculus page</a> di Universitas California, Davis</li> <li><a rel="nofollow" class="external text" href="http://www.math.temple.edu/~cow/">COW: Calculus on the Web</a> di Universitas Temple</li> <li><a rel="nofollow" class="external text" href="http://integrals.wolfram.com/">Online Integrator (WebMathematica)</a> dari Wolfram Research</li> <li><a rel="nofollow" class="external text" href="http://www.ericdigests.org/pre-9217/calculus.htm">The Role of Calculus in College Mathematics</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210726234750/http://www.ericdigests.org/pre-9217/calculus.htm">Diarsipkan</a> 2021-07-26 di <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. dari ERICDigests.org</li> <li><a rel="nofollow" class="external text" href="http://ocw.mit.edu/OcwWeb/Mathematics/index.htm">OpenCourseWare Calculus</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100505005607/http://ocw.mit.edu/OcwWeb/Mathematics/index.htm">Diarsipkan</a> 2010-05-05 di <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. dari <a href="/wiki/Institut_Teknologi_Massachusetts" title="Institut Teknologi Massachusetts">Institut Teknologi Massachusetts</a></li> <li><a rel="nofollow" class="external text" href="http://eom.springer.de/I/i050950.htm">Infinitesimal Calculus</a> Encyclopaedia of Mathematics, Michiel Hazewinkel ed. .</li></ul> <div class="navbox-styles"><style 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