Multivariable Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 0-53849-787-4
ISBN 13: 978-0-53849-787-9

Chapter 15 - Multiple Integrals - 15.3 Exercises - Page 1019: 4

Answer

$$\int_{0}^{2} \int_{y}^{2 y} x y d x d y=6$$

Work Step by Step

Given $$\int_{0}^{2} \int_{y}^{2 y} x y d x d y$$ So, we have \begin{aligned}I&=\int_{0}^{2} \int_{y}^{2 y} x y d x d y\\&=\int_{0}^{2}\left[\frac{x^{2} y}{2}\right]_{y}^{2 y} d y\\ &=\int_{0}^{2}\left[\frac{(2 y)^{2} y}{2}-\frac{(y)^{2} y}{2}\right] d y\\&=\int_{0}^{2} \frac{3 y^{3}}{2} d y\\ &=\left[\frac{3 y^{4}}{8}\right]_{0}^{2}\\ &=\frac{3(2)^{4}}{8}-0\\ &=6 \end{aligned}
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