{"id":1129,"date":"2021-06-30T17:01:57","date_gmt":"2021-06-30T17:01:57","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-the-fundamental-theorem-of-calculus\/"},"modified":"2021-11-17T01:39:02","modified_gmt":"2021-11-17T01:39:02","slug":"summary-of-the-fundamental-theorem-of-calculus","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-the-fundamental-theorem-of-calculus\/","title":{"raw":"Summary of the Fundamental Theorem of Calculus","rendered":"Summary of the Fundamental Theorem of Calculus"},"content":{"raw":"<div id=\"fs-id1170571699029\" class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul>\r\n \t<li>The Mean Value Theorem for Integrals states that for a continuous function over a closed interval, there is a value [latex]c[\/latex] such that [latex]f(c)[\/latex] equals the average value of the function. See the Mean Value Theorem for Integrals.<\/li>\r\n \t<li>The Fundamental Theorem of Calculus, Part 1 shows the relationship between the derivative and the integral. See the Fundamental Theorem of Calculus, Part 1.<\/li>\r\n \t<li>The Fundamental Theorem of Calculus, Part 2 is a formula for evaluating a definite integral in terms of an antiderivative of its integrand. The total area under a curve can be found using this formula. See the Fundamental Theorem of Calculus, Part 2.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1170572183839\" class=\"key-equations\">\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1170572183845\">\r\n \t<li><strong>Mean Value Theorem for Integrals<\/strong>\r\nIf [latex]f(x)[\/latex] is continuous over an interval [latex]\\left[a,b\\right],[\/latex] then there is at least one point [latex]c\\in \\left[a,b\\right][\/latex] such that [latex]f(c)=\\frac{1}{b-a}{\\displaystyle\\int }_{a}^{b}f(x)dx.[\/latex]<\/li>\r\n \t<li><strong>Fundamental Theorem of Calculus Part 1<\/strong>\r\nIf [latex]f(x)[\/latex] is continuous over an interval [latex]\\left[a,b\\right],[\/latex] and the function [latex]F(x)[\/latex] is defined by [latex]F(x)={\\displaystyle\\int }_{a}^{x}f(t)dt,[\/latex] then [latex]{F}^{\\prime }(x)=f(x).[\/latex]<\/li>\r\n \t<li><strong>Fundamental Theorem of Calculus Part 2<\/strong>\r\nIf [latex]f[\/latex] is continuous over the interval [latex]\\left[a,b\\right][\/latex] and [latex]F(x)[\/latex] is any antiderivative of [latex]f(x),[\/latex] then [latex]{\\displaystyle\\int }_{a}^{b}f(x)dx=F(b)-F(a).[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1170572379108\" class=\"definition\">\r\n \t<dt>fundamental theorem of calculus<\/dt>\r\n \t<dd id=\"fs-id1170572379113\">the theorem, central to the entire development of calculus, that establishes the relationship between differentiation and integration<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572379119\" class=\"definition\">\r\n \t<dt>fundamental theorem of calculus, part 1<\/dt>\r\n \t<dd id=\"fs-id1170572379124\">uses a definite integral to define an antiderivative of a function<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572379128\" class=\"definition\">\r\n \t<dt>fundamental theorem of calculus, part 2<\/dt>\r\n \t<dd id=\"fs-id1170572379134\">(also, <strong>evaluation theorem<\/strong>) we can evaluate a definite integral by evaluating the antiderivative of the integrand at the endpoints of the interval and subtracting<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572379144\" class=\"definition\">\r\n \t<dt>mean value theorem for integrals<\/dt>\r\n \t<dd id=\"fs-id1170572379150\">guarantees that a point [latex]c[\/latex] exists such that [latex]f(c)[\/latex] is equal to the average value of the function<\/dd>\r\n<\/dl>","rendered":"<div id=\"fs-id1170571699029\" class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul>\n<li>The Mean Value Theorem for Integrals states that for a continuous function over a closed interval, there is a value [latex]c[\/latex] such that [latex]f(c)[\/latex] equals the average value of the function. See the Mean Value Theorem for Integrals.<\/li>\n<li>The Fundamental Theorem of Calculus, Part 1 shows the relationship between the derivative and the integral. See the Fundamental Theorem of Calculus, Part 1.<\/li>\n<li>The Fundamental Theorem of Calculus, Part 2 is a formula for evaluating a definite integral in terms of an antiderivative of its integrand. The total area under a curve can be found using this formula. See the Fundamental Theorem of Calculus, Part 2.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1170572183839\" class=\"key-equations\">\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1170572183845\">\n<li><strong>Mean Value Theorem for Integrals<\/strong><br \/>\nIf [latex]f(x)[\/latex] is continuous over an interval [latex]\\left[a,b\\right],[\/latex] then there is at least one point [latex]c\\in \\left[a,b\\right][\/latex] such that [latex]f(c)=\\frac{1}{b-a}{\\displaystyle\\int }_{a}^{b}f(x)dx.[\/latex]<\/li>\n<li><strong>Fundamental Theorem of Calculus Part 1<\/strong><br \/>\nIf [latex]f(x)[\/latex] is continuous over an interval [latex]\\left[a,b\\right],[\/latex] and the function [latex]F(x)[\/latex] is defined by [latex]F(x)={\\displaystyle\\int }_{a}^{x}f(t)dt,[\/latex] then [latex]{F}^{\\prime }(x)=f(x).[\/latex]<\/li>\n<li><strong>Fundamental Theorem of Calculus Part 2<\/strong><br \/>\nIf [latex]f[\/latex] is continuous over the interval [latex]\\left[a,b\\right][\/latex] and [latex]F(x)[\/latex] is any antiderivative of [latex]f(x),[\/latex] then [latex]{\\displaystyle\\int }_{a}^{b}f(x)dx=F(b)-F(a).[\/latex]<\/li>\n<\/ul>\n<\/div>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1170572379108\" class=\"definition\">\n<dt>fundamental theorem of calculus<\/dt>\n<dd id=\"fs-id1170572379113\">the theorem, central to the entire development of calculus, that establishes the relationship between differentiation and integration<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572379119\" class=\"definition\">\n<dt>fundamental theorem of calculus, part 1<\/dt>\n<dd id=\"fs-id1170572379124\">uses a definite integral to define an antiderivative of a function<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572379128\" class=\"definition\">\n<dt>fundamental theorem of calculus, part 2<\/dt>\n<dd id=\"fs-id1170572379134\">(also, <strong>evaluation theorem<\/strong>) we can evaluate a definite integral by evaluating the antiderivative of the integrand at the endpoints of the interval and subtracting<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572379144\" class=\"definition\">\n<dt>mean value theorem for integrals<\/dt>\n<dd id=\"fs-id1170572379150\">guarantees that a point [latex]c[\/latex] exists such that [latex]f(c)[\/latex] is equal to the average value of the function<\/dd>\n<\/dl>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1129\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 2. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":16,"template":"","meta":{"_candela_citation":"{\"1\":{\"type\":\"cc\",\"description\":\"Calculus Volume 2\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\"}}","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1129","chapter","type-chapter","status-publish","hentry"],"part":1113,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1129","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1129\/revisions"}],"predecessor-version":[{"id":2468,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1129\/revisions\/2468"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/1113"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1129\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=1129"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=1129"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=1129"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=1129"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}