We saw in Module 1: Functions and Graphs that an even function is a function in which [latex]f(\text{−}x)=f(x)[/latex] for all [latex]x[/latex] in the domain—that is, the graph of the curve is unchanged when [latex]x[/latex] is replaced with −[latex]x[/latex]. The graphs of even functions are symmetric about the [latex]y[/latex]-axis. An odd function is one in which [latex]f(\text{−}x)=\text{−}f(x)[/latex] for all [latex]x[/latex] in the domain, and the graph of the function is symmetric about the origin.

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{−}a,a\right],[/latex] evaluate to zero because the areas above and below the [latex]x[/latex]-axis are equal.

Example: Integrating an Even Function

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.

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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

[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]

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.

[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]

Since [latex]2·\frac{500}{3}=\frac{1000}{3},[/latex] we have verified the formula for even functions in this particular example.

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.

Watch the following video to see the worked solution to Example: Integrating an Even Function.

Example: Integrating an Odd Function

Evaluate the definite integral of the odd function [latex]-5 \sin x[/latex] over the interval [latex]\left[\text{−}\pi ,\pi \right].[/latex]

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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{−}\pi ,0\right],[/latex] and the negative area below the [latex]x[/latex]-axis over [latex]\left[0,\pi \right].[/latex] We have

[latex]\begin{array}{ll}{\int }_{\text{−}\pi }^{\pi }-5 \sin xdx\hfill & =-5(\text{−} \cos x){|}_{\text{−}\pi }^{\pi }\hfill \\ \\ \\ & =5 \cos x{|}_{\text{−}\pi }^{\pi }\hfill \\ & =\left[5 \cos \pi \right]-\left[5 \cos (\text{−}\pi )\right]\hfill \\ & =-5-(-5)\hfill \\ & =0.\hfill \end{array}[/latex]

Figure 4. The graph shows areas between a curve and the x-axis for an odd function.

Watch the following video to see the worked solution to Example: Integrating an Odd Function.