Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 15: Multiple Integrals - Section 15.7 - Triple Integrals in Cylindrical and Spherical Coordinates - Exercises 15.7 - Page 919: 21

Answer

$\pi^2$

Work Step by Step

$\int^{\pi}_0 \int^{\pi}_0 \int^{2sin \phi}_0 p^2 sin\phi $ $ dp $ $ d\phi $ $ d\theta $ =$\frac{8}{3}\int^{\pi}_0\int^{\pi}_0sin^4\phi $ d $\phi $ $ d\theta $ =$\frac{8}{3}\int^{\pi}_0 ([-\frac{sin^3\phi cos\phi}{4}]^\pi_0+\frac{3}{4}\int^{\pi}_0 sin^2 \phi )$ d $\phi $ =$2\int^{\pi}_0 \int^{\pi}_0 sin^2 \phi $ $ d\phi $ $ d\theta $ =$\int^{\pi}_0 [\theta-\frac{sin2\theta}{2}]^\pi_0 d\theta $ =$\int^{\pi}_0 \pi d\theta $ =$\pi^2$
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