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 921: 49

Answer

$\frac{2\pi a^3}{3}$

Work Step by Step

$ V=\int^{2\pi}_0 \int^{2\pi/3}_{\pi/3} \int^a_0 p^2 \sin\phi $ dp $ d\phi $ $ d\theta $ =$\int^{2\pi}_0 \int^{2\pi/3}_{\pi/3} \frac{a^3}{3}\sin \phi d\phi d\theta $ =$\frac{a^3}{3}\int^{2\pi}_0[-cos\phi]^{2\pi/3}_{\pi/3}d\theta $ =$\frac{a^3}{3} \int^{2\pi}_0(\frac{1}{2}+\frac{1}{2})d\theta $ =$\frac{2\pi a^3}{3}$
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