Example 7: Area Under A Semicircle

Find .

The function as given is only defined on the closed interval [-1,1]. In order to use Theorem 2, we extend it to the open interval J = (-∞, ∞) by

Let u = 1 - x². Then du = -2x dx, dx = -du/2x. Δt x = 0, u = 1. Δt x = 1, u = 0. Therefore

Figure 4.4.4

We see in Figure 4.4.4 that as x increases from 0 to 1, u decreases from 1 to 0, so the limits become reversed. The areas shown in Figure 4.4.5 are equal.

Figure 4.4.5