Chapter 5 | Integration
573
Example 5.28 Integrating an Even Function
Integrate the even function ∫ −2 2 ⎛
⎝ 3 x 8 −2 ⎞
⎠ dx and verify that the integration formula for even functions holds.
Solution The symmetry appears in the graphs in Figure 5.35 . Graph (a) shows the region below the curve and above the x -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 x -axis. The signed area of this region is negative. Both views illustrate the symmetry about the y -axis of an even function. We have ∫ −2 2 ⎛ ⎝ 3 x 8 −2 ⎞ ⎠ dx = ⎛ ⎝ x 9 3 −2 x ⎞ ⎠ | −2 2 = (2) (−2)
⎡ ⎣ ⎢ (2) 9 ⎛ ⎝ 512
⎤ ⎦ ⎥ −
⎡ ⎣ ⎢ (−2) 9
⎤ ⎦ ⎥
3 −2
3 −2
⎞ ⎠ −
⎛ ⎝ − 512 3 +4 ⎞ ⎠
=
3 −4
= 1000 3 . 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. ∫ 0 2 ⎛ ⎝ 3 x 8 −2 ⎞ ⎠ dx = ⎛ ⎝ x 9 3 −2 x ⎞ ⎠ | 0 2 = 512 3 −4 = 500 3 Since 2· 500 3 = 1000 3 , we have verified the formula for even functions in this particular example.
Figure 5.35 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.
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