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How To Solve Multiple Integral Problems

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

Step 1

Sketch the problem.

Step 2

Find the inequalities describing the region D.

Step 3

Set up the iterated integral and evaluate.

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,

12_multiple_integrals-94.gif

Figure 12.2.5

12_multiple_integrals-95.gif

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).

12_multiple_integrals-96.gif

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

12_multiple_integrals-97.gif

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

Figure 12.2.6

Example 5


Last Update: 2006-11-05