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 920: 35

Answer

$\frac{8\pi}{3}$

Work Step by Step

$ V=\int^{2\pi}_0 \int^\pi_0 \int^{1-cos\phi}_0 p^2\sin\phi dpd\phi d\theta $ =$\frac{1}{3}\int^{2\pi}_0 \int^\pi_0 (1-cos\phi)^3 \sin \phi $ $ d\phi $ $ d\theta $ =$\frac{1}{3}\int^{2\pi}_0 [\frac{(1-cos\phi)^4}{4}]^\pi_0 d\theta $ =$\frac{1}{12}(2)^4\int^{2\pi}_0 d\theta $ =$\frac{4}{3}(2\pi)$ =$\frac{8\pi}{3}$
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