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Example 8

Find

04_integration-275.gif

Let u = x - x3. Then du = (1 - 3x2) dx. When x = 0, u = 0 - 03 = 0. When x = 1, u = 1 - 13 = 0. Then

04_integration-276.gif

As x goes from 0 to 1, u starts at 0, increases for a time, then drops back to 0 (Figure 4.4.7).

04_integration-280.gif

Figure 4.4.7

We do not know how to find the indefinite integrals in this example. Nevertheless the answer is 0 because on changing variables both limits of integration become the same. Using the Addition Property, we can also see that, for instance,

04_integration-278.gif


Last Update: 2006-11-10