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Iterated Integral Theorem

ITERATED INTEGRAL THEOREM

Let D be a region

a1 ≤ x ≤ a2, b1x ≤ y ≤ b2(x).

The double integral over D is equal to the iterated integral:

12_multiple_integrals-72.gif

Discussion

For a fixed x0, ∫ f(x0, y) dy is the area of the cross section shown in Figure 12.2.1. The Iterated Integral Theorem states that the volume is equal to the integral of the areas of the cross sections.

The proof of the Iterated Integral Theorem is given at the end of this section. When using iterated integrals we must be sure that:

(1)    a1 ≤ a2 and b1(x) ≤ b2(x).

(2)    The differentials dx and dy appear in the right order.

(3)    The outer integral sign has constant limits.

12_multiple_integrals-73.gif

Figure 12.2.1
Home Multiple Integrals Iterated Integrals Iterated Integral Theorem

Last Update: 2006-11-05