Example 2

Evaluate

where D₂ is the region in Figure 12.2.2:

0 ≤ x ≤ 1, x² ≤ y ≤ √x.

The limits on the outside integral are given by 0 ≤ x ≤ 1, and those on the inside integral by x² ≤ y ≤ √x, so the iterated integral is

Figure 12.2.2

The Riemann sums in Section 12.1 were 0.05375, 0.04881.

In many applications the region D is given verbally, and part of the problem is to find inequalities which describe D.