Step 1

Sketch the solid and the region D, as in Figure 12.2.4.

Figure 12.2.4

Step 2

Find the inequalities describing the region D.

This is the hardest step, and gives us the limits of integration. The surfaces z = 0 and z = y - x² intersect at the curve y = x². We see from the figure that D is the region between the curves y = x² and y = 1, so D is given by

-1 ≤ x ≤ 1, x² ≤ y ≤ 1.

Step 3

Set up the iterated integral and evaluate it.