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Product of integrals Suppose f(x, y) = g(x)h(y), where g
Chapter 11, Problem 50AE(choose chapter or problem)
Product of integrals Suppose f(x, y) = g(x)h(y), where g and h are continuous functions for all real values.
a. Show that \(\int_{c}^{d} \int_{a}^{b} f(x, y) \ d x \ d y=\left(\int_{a}^{b} g(x) d x\right)\left(\int_{c}^{d} h(y) \ d y\right)\). Interpret this result geometrically.
b. Write \(\left(\int_{a}^{b} g(x) \ d x\right)^{2}\) as an iterated integral.
c. Use the result of part (a) to evaluate \(\int_{0}^{2 \pi} \int_{10}^{30}(\cos x) e^{-4 y^{2}} \ d y \ d x\)
Questions & Answers
QUESTION:
Product of integrals Suppose f(x, y) = g(x)h(y), where g and h are continuous functions for all real values.
a. Show that \(\int_{c}^{d} \int_{a}^{b} f(x, y) \ d x \ d y=\left(\int_{a}^{b} g(x) d x\right)\left(\int_{c}^{d} h(y) \ d y\right)\). Interpret this result geometrically.
b. Write \(\left(\int_{a}^{b} g(x) \ d x\right)^{2}\) as an iterated integral.
c. Use the result of part (a) to evaluate \(\int_{0}^{2 \pi} \int_{10}^{30}(\cos x) e^{-4 y^{2}} \ d y \ d x\)
ANSWER:Solution 50AE
(a)
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