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 - Review - Exercises - Page 1074: 23

Answer

$\frac{81}{2}$

Work Step by Step

$\int\int_{E}xydV=\int_{0}^{3}\int_{0} ^{x}\int_{0} ^{x+y} xydzdydx$ $=\int_{0}^{3}\int_{0} ^{x} xyz|_{0} ^{x+y}dydx$ $=\int_{0}^{3}\int_{0} ^{x} x^{2}y+xy^{2}dydx$ $=\int_{0}^{3}\frac{ x^{2}y^{2}}{2}+\frac{xy^{3}}{3}|_{0} ^{x}dx$ $=\int_{0}^{3}\frac{ 5x^{4}}{6}dx$ $=[\frac{ x^{5}}{6}]|_{0}^{3}$ $=\frac{81}{2}$
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