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: 37

Answer

$\frac{\pi}{3}$

Work Step by Step

$ V=\int^{2\pi}_0 \int^{\pi/2}_{\pi/4} \int^{2cos\phi}_0 p^2\sin \phi dp d\theta $ =$\frac{8}{3}\int^{2\pi}_0 \int^{\pi/2}_{\pi/4} cos^3\phi \sin \phi d\phi d\theta $ =$\frac{8}{3} \int^{2\pi}_0 [-\frac{cos^4\phi}{4}]^{\pi/2}_{\pi/4}d\theta $ =$(\frac{8}{3})(\frac{1}{16})\int^{2\pi}_0 d\theta $ =$\frac{1}{6}(2\pi)$ =$\frac{\pi}{3}$
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