{"id":1125,"date":"2021-06-30T17:01:56","date_gmt":"2021-06-30T17:01:56","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-the-definite-integral\/"},"modified":"2021-11-17T01:37:43","modified_gmt":"2021-11-17T01:37:43","slug":"summary-of-the-definite-integral","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-the-definite-integral\/","title":{"raw":"Summary of the Definite Integral","rendered":"Summary of the Definite Integral"},"content":{"raw":"<div id=\"fs-id1170572601256\" class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1170572601263\">\r\n \t<li>The definite integral can be used to calculate net signed area, which is the area above the [latex]x[\/latex]-axis minus the area below the [latex]x[\/latex]-axis. Net signed area can be positive, negative, or zero.<\/li>\r\n \t<li>The component parts of the definite integral are the integrand, the variable of integration, and the limits of integration.<\/li>\r\n \t<li>Continuous functions on a closed interval are integrable. Functions that are not continuous may still be integrable, depending on the nature of the discontinuities.<\/li>\r\n \t<li>The properties of definite integrals can be used to evaluate integrals.<\/li>\r\n \t<li>The area under the curve of many functions can be calculated using geometric formulas.<\/li>\r\n \t<li>The average value of a function can be calculated using definite integrals.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1170572601305\" class=\"key-equations\">\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1170572601312\">\r\n \t<li><strong>Definite Integral<\/strong>\r\n[latex]\\displaystyle\\int_a^b f(x) dx = \\underset{n\\to \\infty}{\\lim}\\underset{i=1}{\\overset{n}{\\Sigma}} f(x_i^*) \\Delta x[\/latex]<\/li>\r\n \t<li><strong>Properties of the Definite Integral<\/strong>\r\n[latex]\\displaystyle\\int_a^a f(x) dx = 0[\/latex]\r\n[latex]\\displaystyle\\int_b^a f(x) dx = \u2212\\displaystyle\\int_a^b f(x) dx[\/latex]\r\n[latex]\\displaystyle\\int_a^b [f(x)+g(x)] dx = \\displaystyle\\int_a^b f(x) dx + \\displaystyle\\int_a^b g(x) dx[\/latex]\r\n[latex]\\displaystyle\\int_a^b [f(x)-g(x)] dx = \\displaystyle\\int_a^b f(x) dx - \\displaystyle\\int_a^b g(x) dx[\/latex]\r\n[latex]\\displaystyle\\int_a^b cf(x) dx = c \\displaystyle\\int_a^b f(x) dx[\/latex] for constant [latex]c[\/latex]\r\n[latex]\\displaystyle\\int_a^b f(x) dx = \\displaystyle\\int_a^c f(x) dx + \\displaystyle\\int_c^b f(x) dx[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1170572398858\" class=\"definition\">\r\n \t<dt>average value of a function<\/dt>\r\n \t<dd id=\"fs-id1170572398864\">(or <strong>[latex]f_{\\text{ave}}[\/latex]<\/strong>) the average value of a function on an interval can be found by calculating the definite integral of the function and dividing that value by the length of the interval<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572398877\" class=\"definition\">\r\n \t<dt>definite integral<\/dt>\r\n \t<dd id=\"fs-id1170572398882\">a primary operation of calculus; the area between the curve and the [latex]x[\/latex]-axis over a given interval is a definite integral<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572398892\" class=\"definition\">\r\n \t<dt>integrable function<\/dt>\r\n \t<dd id=\"fs-id1170572398898\">a function is integrable if the limit defining the integral exists; in other words, if the limit of the Riemann sums as [latex]n[\/latex] goes to infinity exists<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572398909\" class=\"definition\">\r\n \t<dt>integrand<\/dt>\r\n \t<dd id=\"fs-id1170572398914\">the function to the right of the integration symbol; the integrand includes the function being integrated<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572398919\" class=\"definition\">\r\n \t<dt>limits of integration<\/dt>\r\n \t<dd id=\"fs-id1170572398925\">these values appear near the top and bottom of the integral sign and define the interval over which the function should be integrated<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572398930\" class=\"definition\">\r\n \t<dt>net signed area<\/dt>\r\n \t<dd id=\"fs-id1170572398936\">the area between a function and the [latex]x[\/latex]-axis such that the area below the [latex]x[\/latex]-axis is subtracted from the area above the [latex]x[\/latex]-axis; the result is the same as the definite integral of the function<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572398956\" class=\"definition\">\r\n \t<dt>total area<\/dt>\r\n \t<dd id=\"fs-id1170572398962\">total area between a function and the [latex]x[\/latex]-axis is calculated by adding the area above the [latex]x[\/latex]-axis and the area below the [latex]x[\/latex]-axis; the result is the same as the definite integral of the absolute value of the function<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571613095\" class=\"definition\">\r\n \t<dt>variable of integration<\/dt>\r\n \t<dd id=\"fs-id1170571613101\">indicates which variable you are integrating with respect to; if it is [latex]x[\/latex], then the function in the integrand is followed by [latex]dx[\/latex]<\/dd>\r\n<\/dl>","rendered":"<div id=\"fs-id1170572601256\" class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1170572601263\">\n<li>The definite integral can be used to calculate net signed area, which is the area above the [latex]x[\/latex]-axis minus the area below the [latex]x[\/latex]-axis. Net signed area can be positive, negative, or zero.<\/li>\n<li>The component parts of the definite integral are the integrand, the variable of integration, and the limits of integration.<\/li>\n<li>Continuous functions on a closed interval are integrable. Functions that are not continuous may still be integrable, depending on the nature of the discontinuities.<\/li>\n<li>The properties of definite integrals can be used to evaluate integrals.<\/li>\n<li>The area under the curve of many functions can be calculated using geometric formulas.<\/li>\n<li>The average value of a function can be calculated using definite integrals.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1170572601305\" class=\"key-equations\">\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1170572601312\">\n<li><strong>Definite Integral<\/strong><br \/>\n[latex]\\displaystyle\\int_a^b f(x) dx = \\underset{n\\to \\infty}{\\lim}\\underset{i=1}{\\overset{n}{\\Sigma}} f(x_i^*) \\Delta x[\/latex]<\/li>\n<li><strong>Properties of the Definite Integral<\/strong><br \/>\n[latex]\\displaystyle\\int_a^a f(x) dx = 0[\/latex]<br \/>\n[latex]\\displaystyle\\int_b^a f(x) dx = \u2212\\displaystyle\\int_a^b f(x) dx[\/latex]<br \/>\n[latex]\\displaystyle\\int_a^b [f(x)+g(x)] dx = \\displaystyle\\int_a^b f(x) dx + \\displaystyle\\int_a^b g(x) dx[\/latex]<br \/>\n[latex]\\displaystyle\\int_a^b [f(x)-g(x)] dx = \\displaystyle\\int_a^b f(x) dx - \\displaystyle\\int_a^b g(x) dx[\/latex]<br \/>\n[latex]\\displaystyle\\int_a^b cf(x) dx = c \\displaystyle\\int_a^b f(x) dx[\/latex] for constant [latex]c[\/latex]<br \/>\n[latex]\\displaystyle\\int_a^b f(x) dx = \\displaystyle\\int_a^c f(x) dx + \\displaystyle\\int_c^b f(x) dx[\/latex]<\/li>\n<\/ul>\n<\/div>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1170572398858\" class=\"definition\">\n<dt>average value of a function<\/dt>\n<dd id=\"fs-id1170572398864\">(or <strong>[latex]f_{\\text{ave}}[\/latex]<\/strong>) the average value of a function on an interval can be found by calculating the definite integral of the function and dividing that value by the length of the interval<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572398877\" class=\"definition\">\n<dt>definite integral<\/dt>\n<dd id=\"fs-id1170572398882\">a primary operation of calculus; the area between the curve and the [latex]x[\/latex]-axis over a given interval is a definite integral<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572398892\" class=\"definition\">\n<dt>integrable function<\/dt>\n<dd id=\"fs-id1170572398898\">a function is integrable if the limit defining the integral exists; in other words, if the limit of the Riemann sums as [latex]n[\/latex] goes to infinity exists<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572398909\" class=\"definition\">\n<dt>integrand<\/dt>\n<dd id=\"fs-id1170572398914\">the function to the right of the integration symbol; the integrand includes the function being integrated<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572398919\" class=\"definition\">\n<dt>limits of integration<\/dt>\n<dd id=\"fs-id1170572398925\">these values appear near the top and bottom of the integral sign and define the interval over which the function should be integrated<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572398930\" class=\"definition\">\n<dt>net signed area<\/dt>\n<dd id=\"fs-id1170572398936\">the area between a function and the [latex]x[\/latex]-axis such that the area below the [latex]x[\/latex]-axis is subtracted from the area above the [latex]x[\/latex]-axis; the result is the same as the definite integral of the function<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572398956\" class=\"definition\">\n<dt>total area<\/dt>\n<dd id=\"fs-id1170572398962\">total area between a function and the [latex]x[\/latex]-axis is calculated by adding the area above the [latex]x[\/latex]-axis and the area below the [latex]x[\/latex]-axis; the result is the same as the definite integral of the absolute value of the function<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571613095\" class=\"definition\">\n<dt>variable of integration<\/dt>\n<dd id=\"fs-id1170571613101\">indicates which variable you are integrating with respect to; if it is [latex]x[\/latex], then the function in the integrand is followed by [latex]dx[\/latex]<\/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-1125\">\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":12,"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-1125","chapter","type-chapter","status-publish","hentry"],"part":1113,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1125","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\/1125\/revisions"}],"predecessor-version":[{"id":2466,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1125\/revisions\/2466"}],"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\/1125\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=1125"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=1125"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=1125"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=1125"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}