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Iterated Integral Theorem
ITERATED INTEGRAL THEOREM Let D be a region a_{1} ≤ x ≤ a_{2}, b_{1}x ≤ y ≤ b_{2}(x). The double integral over D is equal to the iterated integral: Discussion For a fixed x_{0}, ∫ f(x_{0}, 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) a_{1} ≤ a_{2} and b_{1}(x) ≤ b_{2}(x). (2) The differentials dx and dy appear in the right order. (3) The outer integral sign has constant limits. Figure 12.2.1


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