Example 1

Evaluate

where D₁ is the unit square

0 ≤ x ≤ 1, 0≤ y ≤ 1.

The limits of the outside integral are given by 0 ≤ x ≤ 1, and those of the inside integral are given by 0 ≤ y ≤ 1. The iterated integral is thus

The inside integral is

Then

Since D₁ is a rectangle we may also integrate in the other order, and should get the same answer.

The Riemann sums in Section 12.1 were 0.0875, 0.12825.