How To Solve Multiple Integral Problems

Multiple integration problems can be solved by a three-step process as shown in Examples 3 and 4.

We can also integrate over a region in the (y, x) plane instead of the (x, y) plane. A region D in the (y, x) plane has the form

b₁ ≤ y ≤ b₂, a₁ (y) ≤ x ≤ a₂(y),

as shown in Figure 12.2.5.

The double integral over D is equal to the iterated integral with dy on the outside and dx inside,

Figure 12.2.5

Some regions, such as rectangles and ellipses, may be regarded as regions in either the (x, y) plane or the (y, x) plane (Figure 12.2.6).

(a) D as an (x, y) Region

(b)D as a (y, x) Region

Figure 12.2.6

Example 5