{"id":394,"date":"2025-02-13T19:44:34","date_gmt":"2025-02-13T19:44:34","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/integration-formulas-and-the-net-change-theorem-learn-it-3\/"},"modified":"2025-02-13T19:44:34","modified_gmt":"2025-02-13T19:44:34","slug":"integration-formulas-and-the-net-change-theorem-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/integration-formulas-and-the-net-change-theorem-learn-it-3\/","title":{"raw":"Integration Formulas and the Net Change Theorem: Learn It 3","rendered":"Integration Formulas and the Net Change Theorem: Learn It 3"},"content":{"raw":"\n

Integrating Even and Odd Functions<\/h2>\n

Recall that an even function<\/span> is a function in which [latex]f(\\text{\u2212}x)=f(x)[\/latex] for all [latex]x[\/latex] in the domain. This means 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 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

Integrals of even functions, when the limits of integration are from [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

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Integrals of Even and Odd Functions<\/h3>\n

For continuous even functions such that [latex]f(\\text{\u2212}x)=f(x),[\/latex]<\/p>\n

[latex]{\\displaystyle\\int }_{\\text{\u2212}a}^{a}f(x)dx=2{\\displaystyle\\int }_{0}^{a}f(x)dx.[\/latex]<\/div>\n
 <\/div>\n

For continuous odd functions such that [latex]f(\\text{\u2212}x)=\\text{\u2212}f(x),[\/latex]<\/p>\n

[latex]{\\displaystyle\\int }_{\\text{\u2212}a}^{a}f(x)dx=0.[\/latex]<\/div>\n<\/div>\n<\/section>\n
\n

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\n[reveal-answer q=\"fs-id1170572551794\"]Show Solution[\/reveal-answer]
\n[hidden-answer a=\"fs-id1170572551794\"]\n\n

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.<\/p>\n\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]\"Two 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]\n\n

We have,<\/p>\n

[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

To verify the integration formula for even functions, we can calculate the integral from [latex]0[\/latex] to [latex]2[\/latex] and double it, then check to make sure we get the same answer.<\/p>\n

[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

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

Watch the following video to see the worked solution to this example.<\/p>\n