{"id":1132,"date":"2021-06-30T17:01:57","date_gmt":"2021-06-30T17:01:57","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/integrating-even-and-odd-functions\/"},"modified":"2022-03-19T03:20:32","modified_gmt":"2022-03-19T03:20:32","slug":"integrating-even-and-odd-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/integrating-even-and-odd-functions\/","title":{"raw":"Integrating Even and Odd Functions","rendered":"Integrating Even and Odd Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Apply the integrals of odd and even functions<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1170571711325\">We saw in Module 1: Functions and Graphs that an <span class=\"no-emphasis\">even function<\/span> is a function in which [latex]f(\\text{\u2212}x)=f(x)[\/latex] for all [latex]x[\/latex] in the domain\u2014that is, the graph of the curve is unchanged when [latex]x[\/latex] is replaced with \u2212[latex]x[\/latex]. The graphs of even functions are symmetric about the [latex]y[\/latex]-axis. An <span class=\"no-emphasis\">odd function<\/span> is one in which [latex]f(\\text{\u2212}x)=\\text{\u2212}f(x)[\/latex] for all [latex]x[\/latex] in the domain, and the graph of the function is symmetric about the origin.<\/p>\r\n<p id=\"fs-id1170572503213\">Integrals of even functions, when the limits of integration are from \u2212[latex]a[\/latex] to [latex]a[\/latex], involve two equal areas, because they are symmetric about the [latex]y[\/latex]-axis. Integrals of odd functions, when the limits of integration are similarly [latex]\\left[\\text{\u2212}a,a\\right],[\/latex] evaluate to zero because the areas above and below the [latex]x[\/latex]-axis are equal.<\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Integrals of Even and Odd Functions<\/h3>\r\n\r\n<hr \/>\r\n\r\nFor continuous even functions such that [latex]f(\\text{\u2212}x)=f(x),[\/latex]\r\n<div id=\"fs-id1170572587715\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]{\\displaystyle\\int }_{\\text{\u2212}a}^{a}f(x)dx=2{\\displaystyle\\int }_{0}^{a}f(x)dx.[\/latex]<\/div>\r\n<div><\/div>\r\n<p id=\"fs-id1170572380029\">For continuous odd functions such that [latex]f(\\text{\u2212}x)=\\text{\u2212}f(x),[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1170572380064\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]{\\displaystyle\\int }_{\\text{\u2212}a}^{a}f(x)dx=0.[\/latex]<\/div>\r\n<\/div>\r\n<div>It may be useful to recall how to quickly determine whether a function is even, odd or neither.<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Recall: How to determine whether a function is even, odd or neither<\/h3>\r\n<p id=\"fs-id1170572169681\">Determine whether each of the following functions is even, odd, or neither.<\/p>\r\n\r\n<ol id=\"fs-id1170572169684\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]f(x)=-5x^4+7x^2-2[\/latex]<\/li>\r\n \t<li>[latex]f(x)=2x^5-4x+5[\/latex]<\/li>\r\n \t<li>[latex]f(x)=\\large{\\frac{3x}{x^2+1}}[\/latex]<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1170572477853\">To determine whether a function is even or odd, we evaluate [latex]f(\u2212x)[\/latex] and compare it to [latex]f(x)[\/latex] and [latex]\u2212f(x)[\/latex].<\/p>\r\n\r\n<ol id=\"fs-id1170572477904\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]f(\u2212x)=-5(\u2212x)^4+7(\u2212x)^2-2=-5x^4+7x^2-2=f(x)[\/latex]. Therefore, [latex]f[\/latex] is even.<\/li>\r\n \t<li>[latex]f(\u2212x)=2(\u2212x)^5-4(\u2212x)+5=-2x^5+4x+5[\/latex]. Now, [latex]f(\u2212x)\\ne f(x)[\/latex]. Furthermore, noting that [latex]\u2212f(x)=-2x^5+4x-5[\/latex], we see that [latex]f(\u2212x)\\ne \u2212f(x)[\/latex]. Therefore, [latex]f[\/latex] is neither even nor odd.<\/li>\r\n \t<li>[latex]f(\u2212x)=\\frac{3(\u2212x)}{((\u2212x)^2+1)}=\\frac{-3x}{(x^2+1)}=\u2212\\left[\\frac{3x}{(x^2+1)}\\right]=\u2212f(x)[\/latex]. Therefore, [latex]f[\/latex] is odd.<\/li>\r\n<\/ol>\r\n&nbsp;\r\n\r\n<\/div>\r\n<div><\/div>\r\n<div id=\"fs-id1170572624845\" class=\"textbox exercises\">\r\n<h3>Example: Integrating an Even Function<\/h3>\r\nIntegrate the even function [latex]{\\displaystyle\\int }_{-2}^{2}(3{x}^{8}-2)dx[\/latex] and verify that the integration formula for even functions holds.\r\n<div id=\"fs-id1170572624847\" class=\"exercise\">[reveal-answer q=\"fs-id1170572551794\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572551794\"]\r\n<p id=\"fs-id1170572551794\">The symmetry appears in the graphs in Figure 3. Graph (a) shows the region below the curve and above the [latex]x[\/latex]-axis. We have to zoom in to this graph by a huge amount to see the region. Graph (b) shows the region above the curve and below the [latex]x[\/latex]-axis. The signed area of this region is negative. Both views illustrate the symmetry about the [latex]y[\/latex]-axis of an even function. We have<\/p>\r\n\r\n<div id=\"fs-id1170572551814\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll}{\\int }_{-2}^{2}(3{x}^{8}-2)dx\\hfill &amp; =(\\frac{{x}^{9}}{3}-2x){|}_{-2}^{2}\\hfill \\\\ \\\\ \\\\ &amp; =\\left[\\frac{{(2)}^{9}}{3}-2(2)\\right]-\\left[\\frac{{(-2)}^{9}}{3}-2(-2)\\right]\\hfill \\\\ &amp; =(\\frac{512}{3}-4)-(-\\frac{512}{3}+4)\\hfill \\\\ &amp; =\\frac{1000}{3}.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1170571775811\">To verify the integration formula for even functions, we can calculate the integral from 0 to 2 and double it, then check to make sure we get the same answer.<\/p>\r\n\r\n<div id=\"fs-id1170572444219\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll}{\\int }_{0}^{2}(3{x}^{8}-2)dx\\hfill &amp; =(\\frac{{x}^{9}}{3}-2x){|}_{0}^{2}\\hfill \\\\ \\\\ &amp; =\\frac{512}{3}-4\\hfill \\\\ &amp; =\\frac{500}{3}\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1170572163872\">Since [latex]2\u00b7\\frac{500}{3}=\\frac{1000}{3},[\/latex] we have verified the formula for even functions in this particular example.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11204158\/CNX_Calc_Figure_05_04_005.jpg\" alt=\"Two graphs of the same function f(x) = 3x^8 \u2013 2, side by side. It is symmetric about the y axis, has x-intercepts at (-1,0) and (1,0), and has a y-intercept at (0,-2). The function decreases rapidly as x increases until about -.5, where it levels off at -2. Then, at about .5, it increases rapidly as a mirror image. The first graph is zoomed-out and shows the positive area between the curve and the x-axis over [-2,-1] and [1,2]. The second is zoomed-in and shows the negative area between the curve and the x-axis over [-1,1].\" width=\"975\" height=\"363\" \/> Figure 3. Graph (a) shows the positive area between the curve and the x-axis, whereas graph (b) shows the negative area between the curve and the x-axis. Both views show the symmetry about the y-axis.[\/caption][\/hidden-answer]<\/div>\r\n<\/div>\r\nWatch the following video to see the worked solution to Example: Integrating an Even Function.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/v7nDnOyx8Mw?controls=0&amp;start=795&amp;end=850&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>[reveal-answer q=\"266834\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266834\"]For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/5.4IntegrationFormulasAndTheNetChangeTheorem795to850_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"5.4 Integration Formulas and the Net Change Theorem\" here (opens in new window)<\/a>.[\/hidden-answer]\r\n<div id=\"fs-id1170571599733\" class=\"textbox exercises\">\r\n<h3>Example: Integrating an Odd Function<\/h3>\r\nEvaluate the definite integral of the odd function [latex]-5 \\sin x[\/latex] over the interval [latex]\\left[\\text{\u2212}\\pi ,\\pi \\right].[\/latex]\r\n\r\n[reveal-answer q=\"5372881\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"5372881\"]\r\n<div id=\"fs-id1170571599735\" class=\"exercise\">\r\n<div class=\"solution\">\r\n<p id=\"fs-id1170572229803\">The graph is shown in Figure 4. We can see the symmetry about the origin by the positive area above the [latex]x[\/latex]-axis over [latex]\\left[\\text{\u2212}\\pi ,0\\right],[\/latex] and the negative area below the [latex]x[\/latex]-axis over [latex]\\left[0,\\pi \\right].[\/latex] We have<\/p>\r\n\r\n<div id=\"fs-id1170572621622\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll}{\\int }_{\\text{\u2212}\\pi }^{\\pi }-5 \\sin xdx\\hfill &amp; =-5(\\text{\u2212} \\cos x){|}_{\\text{\u2212}\\pi }^{\\pi }\\hfill \\\\ \\\\ \\\\ &amp; =5 \\cos x{|}_{\\text{\u2212}\\pi }^{\\pi }\\hfill \\\\ &amp; =\\left[5 \\cos \\pi \\right]-\\left[5 \\cos (\\text{\u2212}\\pi )\\right]\\hfill \\\\ &amp; =-5-(-5)\\hfill \\\\ &amp; =0.\\hfill \\end{array}[\/latex]<\/div>\r\n<div><\/div>\r\n<div>[caption id=\"\" align=\"aligncenter\" width=\"325\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11204202\/CNX_Calc_Figure_05_04_006.jpg\" alt=\"A graph of the given function f(x) = -5 sin(x). The area under the function but above the x-axis is shaded over [-pi, 0], and the area above the function and under the x-axis is shaded over [0, pi].\" width=\"325\" height=\"433\" \/> Figure 4. The graph shows areas between a curve and the x-axis for an odd function.[\/caption][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nWatch the following video to see the worked solution to Example: Integrating an Odd Function.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/v7nDnOyx8Mw?controls=0&amp;start=853&amp;end=917&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>[reveal-answer q=\"266833\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266833\"]For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/5.4IntegrationFormulasAndTheNetChangeTheorem853to917_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"5.4 Integration Formulas and the Net Change Theorem\" here (opens in new window)<\/a>.[\/hidden-answer]\r\n<div id=\"fs-id1170572337805\" class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1170572337813\">Integrate the function [latex]{\\displaystyle\\int }_{-2}^{2}{x}^{4}dx.[\/latex]<\/p>\r\n[reveal-answer q=\"488209\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"488209\"]\r\n<p id=\"fs-id1170572337844\">Integrate an even function.<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1170572337850\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572337850\"]\r\n<p id=\"fs-id1170572337850\">[latex]\\frac{64}{5}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]223802[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Apply the integrals of odd and even functions<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1170571711325\">We saw in Module 1: Functions and Graphs that an <span class=\"no-emphasis\">even function<\/span> is a function in which [latex]f(\\text{\u2212}x)=f(x)[\/latex] for all [latex]x[\/latex] in the domain\u2014that is, the graph of the curve is unchanged when [latex]x[\/latex] is replaced with \u2212[latex]x[\/latex]. The graphs of even functions are symmetric about the [latex]y[\/latex]-axis. An <span class=\"no-emphasis\">odd function<\/span> is one in which [latex]f(\\text{\u2212}x)=\\text{\u2212}f(x)[\/latex] for all [latex]x[\/latex] in the domain, and the graph of the function is symmetric about the origin.<\/p>\n<p id=\"fs-id1170572503213\">Integrals of even functions, when the limits of integration are from \u2212[latex]a[\/latex] to [latex]a[\/latex], involve two equal areas, because they are symmetric about the [latex]y[\/latex]-axis. Integrals of odd functions, when the limits of integration are similarly [latex]\\left[\\text{\u2212}a,a\\right],[\/latex] evaluate to zero because the areas above and below the [latex]x[\/latex]-axis are equal.<\/p>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Integrals of Even and Odd Functions<\/h3>\n<hr \/>\n<p>For continuous even functions such that [latex]f(\\text{\u2212}x)=f(x),[\/latex]<\/p>\n<div id=\"fs-id1170572587715\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]{\\displaystyle\\int }_{\\text{\u2212}a}^{a}f(x)dx=2{\\displaystyle\\int }_{0}^{a}f(x)dx.[\/latex]<\/div>\n<div><\/div>\n<p id=\"fs-id1170572380029\">For continuous odd functions such that [latex]f(\\text{\u2212}x)=\\text{\u2212}f(x),[\/latex]<\/p>\n<div id=\"fs-id1170572380064\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]{\\displaystyle\\int }_{\\text{\u2212}a}^{a}f(x)dx=0.[\/latex]<\/div>\n<\/div>\n<div>It may be useful to recall how to quickly determine whether a function is even, odd or neither.<\/div>\n<div class=\"textbox examples\">\n<h3>Recall: How to determine whether a function is even, odd or neither<\/h3>\n<p id=\"fs-id1170572169681\">Determine whether each of the following functions is even, odd, or neither.<\/p>\n<ol id=\"fs-id1170572169684\" style=\"list-style-type: lower-alpha;\">\n<li>[latex]f(x)=-5x^4+7x^2-2[\/latex]<\/li>\n<li>[latex]f(x)=2x^5-4x+5[\/latex]<\/li>\n<li>[latex]f(x)=\\large{\\frac{3x}{x^2+1}}[\/latex]<\/li>\n<\/ol>\n<p id=\"fs-id1170572477853\">To determine whether a function is even or odd, we evaluate [latex]f(\u2212x)[\/latex] and compare it to [latex]f(x)[\/latex] and [latex]\u2212f(x)[\/latex].<\/p>\n<ol id=\"fs-id1170572477904\" style=\"list-style-type: lower-alpha;\">\n<li>[latex]f(\u2212x)=-5(\u2212x)^4+7(\u2212x)^2-2=-5x^4+7x^2-2=f(x)[\/latex]. Therefore, [latex]f[\/latex] is even.<\/li>\n<li>[latex]f(\u2212x)=2(\u2212x)^5-4(\u2212x)+5=-2x^5+4x+5[\/latex]. Now, [latex]f(\u2212x)\\ne f(x)[\/latex]. Furthermore, noting that [latex]\u2212f(x)=-2x^5+4x-5[\/latex], we see that [latex]f(\u2212x)\\ne \u2212f(x)[\/latex]. Therefore, [latex]f[\/latex] is neither even nor odd.<\/li>\n<li>[latex]f(\u2212x)=\\frac{3(\u2212x)}{((\u2212x)^2+1)}=\\frac{-3x}{(x^2+1)}=\u2212\\left[\\frac{3x}{(x^2+1)}\\right]=\u2212f(x)[\/latex]. Therefore, [latex]f[\/latex] is odd.<\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n<\/div>\n<div><\/div>\n<div id=\"fs-id1170572624845\" class=\"textbox exercises\">\n<h3>Example: Integrating an Even Function<\/h3>\n<p>Integrate the even function [latex]{\\displaystyle\\int }_{-2}^{2}(3{x}^{8}-2)dx[\/latex] and verify that the integration formula for even functions holds.<\/p>\n<div id=\"fs-id1170572624847\" class=\"exercise\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572551794\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572551794\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572551794\">The symmetry appears in the graphs in Figure 3. Graph (a) shows the region below the curve and above the [latex]x[\/latex]-axis. We have to zoom in to this graph by a huge amount to see the region. Graph (b) shows the region above the curve and below the [latex]x[\/latex]-axis. The signed area of this region is negative. Both views illustrate the symmetry about the [latex]y[\/latex]-axis of an even function. We have<\/p>\n<div id=\"fs-id1170572551814\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll}{\\int }_{-2}^{2}(3{x}^{8}-2)dx\\hfill & =(\\frac{{x}^{9}}{3}-2x){|}_{-2}^{2}\\hfill \\\\ \\\\ \\\\ & =\\left[\\frac{{(2)}^{9}}{3}-2(2)\\right]-\\left[\\frac{{(-2)}^{9}}{3}-2(-2)\\right]\\hfill \\\\ & =(\\frac{512}{3}-4)-(-\\frac{512}{3}+4)\\hfill \\\\ & =\\frac{1000}{3}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1170571775811\">To verify the integration formula for even functions, we can calculate the integral from 0 to 2 and double it, then check to make sure we get the same answer.<\/p>\n<div id=\"fs-id1170572444219\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll}{\\int }_{0}^{2}(3{x}^{8}-2)dx\\hfill & =(\\frac{{x}^{9}}{3}-2x){|}_{0}^{2}\\hfill \\\\ \\\\ & =\\frac{512}{3}-4\\hfill \\\\ & =\\frac{500}{3}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1170572163872\">Since [latex]2\u00b7\\frac{500}{3}=\\frac{1000}{3},[\/latex] we have verified the formula for even functions in this particular example.<\/p>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11204158\/CNX_Calc_Figure_05_04_005.jpg\" alt=\"Two graphs of the same function f(x) = 3x^8 \u2013 2, side by side. It is symmetric about the y axis, has x-intercepts at (-1,0) and (1,0), and has a y-intercept at (0,-2). The function decreases rapidly as x increases until about -.5, where it levels off at -2. Then, at about .5, it increases rapidly as a mirror image. The first graph is zoomed-out and shows the positive area between the curve and the x-axis over &#091;-2,-1&#093; and &#091;1,2&#093;. The second is zoomed-in and shows the negative area between the curve and the x-axis over &#091;-1,1&#093;.\" width=\"975\" height=\"363\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 3. Graph (a) shows the positive area between the curve and the x-axis, whereas graph (b) shows the negative area between the curve and the x-axis. Both views show the symmetry about the y-axis.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Integrating an Even Function.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/v7nDnOyx8Mw?controls=0&amp;start=795&amp;end=850&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266834\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266834\" class=\"hidden-answer\" style=\"display: none\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/5.4IntegrationFormulasAndTheNetChangeTheorem795to850_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;5.4 Integration Formulas and the Net Change Theorem&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<div id=\"fs-id1170571599733\" class=\"textbox exercises\">\n<h3>Example: Integrating an Odd Function<\/h3>\n<p>Evaluate the definite integral of the odd function [latex]-5 \\sin x[\/latex] over the interval [latex]\\left[\\text{\u2212}\\pi ,\\pi \\right].[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q5372881\">Show Solution<\/span><\/p>\n<div id=\"q5372881\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1170571599735\" class=\"exercise\">\n<div class=\"solution\">\n<p id=\"fs-id1170572229803\">The graph is shown in Figure 4. We can see the symmetry about the origin by the positive area above the [latex]x[\/latex]-axis over [latex]\\left[\\text{\u2212}\\pi ,0\\right],[\/latex] and the negative area below the [latex]x[\/latex]-axis over [latex]\\left[0,\\pi \\right].[\/latex] We have<\/p>\n<div id=\"fs-id1170572621622\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll}{\\int }_{\\text{\u2212}\\pi }^{\\pi }-5 \\sin xdx\\hfill & =-5(\\text{\u2212} \\cos x){|}_{\\text{\u2212}\\pi }^{\\pi }\\hfill \\\\ \\\\ \\\\ & =5 \\cos x{|}_{\\text{\u2212}\\pi }^{\\pi }\\hfill \\\\ & =\\left[5 \\cos \\pi \\right]-\\left[5 \\cos (\\text{\u2212}\\pi )\\right]\\hfill \\\\ & =-5-(-5)\\hfill \\\\ & =0.\\hfill \\end{array}[\/latex]<\/div>\n<div><\/div>\n<div>\n<div style=\"width: 335px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11204202\/CNX_Calc_Figure_05_04_006.jpg\" alt=\"A graph of the given function f(x) = -5 sin(x). The area under the function but above the x-axis is shaded over &#091;-pi, 0&#093;, and the area above the function and under the x-axis is shaded over &#091;0, pi&#093;.\" width=\"325\" height=\"433\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 4. The graph shows areas between a curve and the x-axis for an odd function.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Integrating an Odd Function.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/v7nDnOyx8Mw?controls=0&amp;start=853&amp;end=917&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266833\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266833\" class=\"hidden-answer\" style=\"display: none\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/5.4IntegrationFormulasAndTheNetChangeTheorem853to917_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;5.4 Integration Formulas and the Net Change Theorem&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<div id=\"fs-id1170572337805\" class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1170572337813\">Integrate the function [latex]{\\displaystyle\\int }_{-2}^{2}{x}^{4}dx.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q488209\">Hint<\/span><\/p>\n<div id=\"q488209\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572337844\">Integrate an even function.<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572337850\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572337850\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572337850\">[latex]\\frac{64}{5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm223802\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=223802&theme=oea&iframe_resize_id=ohm223802&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\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-1132\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>5.4 Integration Formulas and the Net Change Theorem. <strong>Authored by<\/strong>: Ryan Melton. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><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":19,"template":"","meta":{"_candela_citation":"[{\"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\"},{\"type\":\"original\",\"description\":\"5.4 Integration Formulas and the Net Change Theorem\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1132","chapter","type-chapter","status-publish","hentry"],"part":1113,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1132","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":3,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1132\/revisions"}],"predecessor-version":[{"id":2645,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1132\/revisions\/2645"}],"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\/1132\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=1132"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=1132"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=1132"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=1132"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}