{"id":72538,"date":"2021-05-06T13:56:31","date_gmt":"2021-05-06T18:56:31","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=72538"},"modified":"2026-04-23T11:18:15","modified_gmt":"2026-04-23T16:18:15","slug":"second-fundamental-theorem-of-calculus","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/second-fundamental-theorem-of-calculus\/","title":{"rendered":"Second Fundamental Theorem of Calculus"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_Z9Jt4WF6Aew\" style=\"position: relative;\">\n\t\t\t<picture>\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.webp\" type=\"image\/webp\">\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" type=\"image\/jpeg\"> \n\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" loading=\"eager\" id=\"videoThumbnailImage_Z9Jt4WF6Aew\" data-source-videoID=\"Z9Jt4WF6Aew\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Second Fundamental Theorem of Calculus Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Second Fundamental Theorem of Calculus\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_Z9Jt4WF6Aew:hover {cursor:pointer;} img#videoThumbnailImage_Z9Jt4WF6Aew {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/02\/1770-second-fundamental-theorem-of-calculus-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_Z9Jt4WF6Aew\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_Z9Jt4WF6Aew\" style=\"display: none;position: absolute;top: -24px;width: 100%;text-align: center;\"><span style=\"font-style: italic;font-size: small;border-top: 1px solid #fc0;\">Having trouble? <a href=\"https:\/\/www.youtube.com\/watch?v='+videoId+'\" target=\"_blank\">Click here to watch on YouTube.<\/a><\/span><\/div>';\n\t\t\t\tlet tag = document.createElement(\"iframe\");\n\t\t\t\ttag.id = \"yt\" + videoId;\n\t\t\t\ttag.src = \"https:\/\/www.youtube-nocookie.com\/embed\/\" + videoId + \"?autoplay=1&controls=1&wmode=opaque&rel=0&egm=0&iv_load_policy=3&hd=0&enablejsapi=1\";\n\t\t\t\ttag.frameborder = 0;\n\t\t\t\ttag.allow = \"autoplay; fullscreen\";\n\t\t\t\ttag.width = this.width;\n\t\t\t\ttag.height = this.height;\n\t\t\t\ttag.setAttribute(\"data-matomo-title\",\"Second Fundamental Theorem of Calculus\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_Z9Jt4WF6Aew\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_Z9Jt4WF6Aew\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_Z9Jt4WF6Aew\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction XLW_Function() {\n  var x = document.getElementById(\"XLW\");\n  if (x.style.display === \"none\") {\n    x.style.display = \"block\";\n  } else {\n    x.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<div class=\"moc-toc hide-on-desktop hide-on-tablet\">\n<div><button onclick=\"XLW_Function()\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/12\/toc2.svg\" width=\"16\" height=\"16\" alt=\"show or hide table of contents\"><\/button><\/p>\n<p>On this page<\/p>\n<\/div>\n<nav id=\"XLW\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#Statement_and_Meaning_of_the_Theorem\" class=\"smooth-scroll\">Statement and Meaning of the Theorem<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Evaluating_an_Accumulation_Function\" class=\"smooth-scroll\">Evaluating an Accumulation Function<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Differentiating_an_Accumulation_Function\" class=\"smooth-scroll\">Differentiating an Accumulation Function<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Practice_Problems\" class=\"smooth-scroll\">Practice Problems<\/a><\/li>\n<\/ul>\n<\/nav>\n<\/div>\n<div class=\"accordion\"><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>Hello and welcome to this video about the second fundamental theorem of calculus!<\/p>\n<p>We will explore what this part says and how it works, but first a refresher of the first part:<\/p>\n<p>If \\(f(x)\\) is continuous on \\([a,b]\\) and \\(\\int f(x)dx=F(x)\\) on \\([a,b]\\), then:<\/p>\n<div class=\"examplesentence\">\\(\\int_a^bf(x)dx=F(b)-F(a)\\)<\/div>\n<p>\n&nbsp;<br \/>\nLet\u2019s get right to it!<\/p>\n<h2><span id=\"Statement_and_Meaning_of_the_Theorem\" class=\"m-toc-anchor\"><\/span>Statement and Meaning of the Theorem<\/h2>\n<p>\nThe second part of the fundamental theorem of calculus is closely related to the first and says:<\/p>\n<div class=\"yellow-quote\" style=\"font-weight: normal\">If \\(f(x)\\) is is continuous on \\([a,b]\\) and the function \\(F(x)\\) is defined by \\(F(x)=\\int_a^xf(t)dt\\), then \\(F'(x)=f(x)\\) over \\([a,b]\\).<\/div>\n<p>The biggest implication with this part is that it guarantees that every integrable function has an antiderivative. Notice how \\(F(x)\\) is defined as an integral function\u2014\\(x\\) can be any value and the function will return a number. This is what\u2019s called an accumulation function.<\/p>\n<p>Let\u2019s see how this part of the theorem works!<\/p>\n<h2><span id=\"Evaluating_an_Accumulation_Function\" class=\"m-toc-anchor\"><\/span>Evaluating an Accumulation Function<\/h2>\n<p>\nSuppose \\(g(x)=\\int_1^xf(t)\u2009dt\\). Use the given values of \\(F(t)\\) to evaluate \\(g(x)\\).<\/p>\n<ul style=\"list-style-type: none\">\n<li style=\"margin-bottom: 10px\">\\(F(0)=0\\)<\/li>\n<li style=\"margin-bottom: 10px\">\\(F(1) =2\\)<\/li>\n<li style=\"margin-bottom: 10px\">\\(F(2)=-3\\)<\/li>\n<li>\\(F(4)=5\\)<\/li>\n<\/ul>\n<table class=\"ATable\" style=\"margin: auto\">\n<thead>\n<tr>\n<th>\\(x\\)<\/th>\n<th>\\(g(x)\\)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>0<\/td>\n<td>\\(\\int_1^0f(t)\u2009dt=F(0)-F(1)=-2\\)<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>\\(\\int_1^1F(t)\u2009dt=0\\)<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>\\(\\int_1^2f(t)\u2009dt=F(2)-F(1)=-5\\)<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>\\(\\int_1^4f(t)\u2009dt=F(4)-F(1)=3\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\n&nbsp;<br \/>\nLet\u2019s try another one!<\/p>\n<h2><span id=\"Differentiating_an_Accumulation_Function\" class=\"m-toc-anchor\"><\/span>Differentiating an Accumulation Function<\/h2>\n<p>\nIf \\(hx=\\int_2^x(3t^2+4t)dt\\), then, by the theorem, we know \\(h&#8217;x=3x^2+4x\\). Notice the derivative function is expressed in terms of \\(x\\) instead of \\(t\\) since the integral function is a function of \\(x\\).<\/p>\n<p>Why is this true? Well, let\u2019s take a look!<\/p>\n<div class=\"examplesentence\">\n<p style=\"margin-bottom: 14px\">\\(h(x)=\\int_2^x(3t^2+4t)dt\\)<\/p>\n<p style=\"margin-bottom: 14px\">=\\(t^3+2t^2|_2^x\\)<\/p>\n<p style=\"margin-bottom: 14px\">=\\((x^3+2x^2)-(2^3+2(2)^2\\)<\/p>\n<p style=\"margin-bottom: 0px\">=\\(x^3+2x^2-16\\)<\/p>\n<\/div>\n<p>\n&nbsp;<br \/>\nTherefore, \\(h&#8217;x=3x^2+4x\\).<\/p>\n<p>This is especially powerful with more complex antiderivatives like this:<\/p>\n<div class=\"examplesentence\">If \\(r(x)=\\int_3^x\\frac{\\sqrt{t}}{t^2-3}dt\\), then \\(r'(x)=\\frac{\\sqrt{x}}{x^2-3}dx\\).<\/div>\n<p>\n&nbsp;<\/p>\n<h2><span id=\"Practice_Problems\" class=\"m-toc-anchor\"><\/span>Practice Problems<\/h2>\n<p>\nLet\u2019s get a little practice in.<\/p>\n<h3><span id=\"Finding_a_Formula_for_and\" class=\"m-toc-anchor\"><\/span>Finding a Formula for \\(g(x)\\) and \\(g'(x)\\)<\/h3>\n<p>\nIf \\(g(x)=\\int_{-5}^{2x}(t^2-t-3)dt\\), find a formula for \\(g(x)\\) and calculate \\(g'(x)\\).<\/p>\n<div class=\"examplesentence\">\n<p style=\"margin-bottom: 1em\">\\(g(x)=\\frac{1}{3}t^3-\\frac{1}{2}t^2-3t|_{-5}^{2x}\\)<\/p>\n<p class=\"longmath\" style=\"margin-bottom: 1em\">\\(g(x)=(\\frac{8x^3}{3}-\\frac{4x^2}{2}-6x)-(-\\frac{125}{3}-\\frac{25}{2}+15)\\)<\/p>\n<p style=\"margin-bottom: 1em\">\\(g(x)=\\frac{8x^3}{3}-\\frac{4x^2}{2}-6x+\\frac{235}{6}\\)<\/p>\n\\(g'(x)=8x^2-4x-6\\)\n<\/div>\n<p>\n&nbsp;<\/p>\n<h3><span id=\"Using_a_Table_of_Values\" class=\"m-toc-anchor\"><\/span>Using a Table of \\(F(t)\\)-Values<\/h3>\n<p>\nSuppose \\(g(x)=\\int{-2}^{x}f(t)\u2009dt\\). Use the provided values of \\(F(t)\\) to evaluate \\(g(x)\\).<\/p>\n<ul style=\"list-style-type: none\">\n<li style=\"margin-bottom: 10px\">\\(F(-1)=6\\)<\/li>\n<li style=\"margin-bottom: 10px\">\\(F(-5) =0\\)<\/li>\n<li style=\"margin-bottom: 10px\">\\(F(2)=10\\)<\/li>\n<li>\\(F(-2)=4\\)<\/li>\n<\/ul>\n<div class=\"longmath\">\n<table class=\"ATable\" style=\"margin: auto\">\n<thead>\n<tr>\n<th>\\(x\\)<\/th>\n<th>\\(g(x)\\)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>\u22121<\/td>\n<td>\\(\\int_{-2}^{-1}f(t)\u2009dt=F(-1)-F(-2)=6-4=2\\)<\/td>\n<\/tr>\n<tr>\n<td>\u22125<\/td>\n<td>\\(\\int_{-2}^{-5}f(t)\u2009dt=F(-5)-F(-2)=0-4=-4\\)<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>\\(\\int_{-2}^{2}f(t)\u2009dt=F(2)-F(-2)=10-4=6\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>\n&nbsp;<br \/>\nI hope that this video helped with your understanding of the second fundamental theorem of calculus!<\/p>\n<\/div>\n<\/div>\n\n<div class=\"home-buttons\">\n<p><a href=\"https:\/\/www.mometrix.com\/academy\/calculus\/\">Return to Calculus Videos<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Return to Calculus Videos<\/p>\n","protected":false},"author":1,"featured_media":100786,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-72538","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-calculus-videos","7":"page_category-long-math","8":"page_category-video-pages-for-study-course-sidebar-ad","9":"page_type-video","10":"subject_matter-math"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/72538","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/comments?post=72538"}],"version-history":[{"count":6,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/72538\/revisions"}],"predecessor-version":[{"id":292133,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/72538\/revisions\/292133"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/100786"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=72538"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}