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: 23

Answer

$\frac{\pi}{3}$

Work Step by Step

$\int^{2\pi}_0 \int^\pi _0 \int^{(1-cos\phi)/2}_0 p^2 \sin\phi $ $ dp $ $ d\phi $ $ d\theta $ =$\frac{1}{24} \int^{2\pi}_0 \int^{\pi}_0 (1-cos\phi)^3 sin\phi $ $ d\phi $ $ d\theta $ =$\frac{1}{96} \int^{2\pi}_0 [(1-cos\phi)]^\pi_0 d\theta $ =$\frac{1}{96}\int^{2\pi}_0(2^4-0)d\theta $ =$\frac{16}{96} \int^{2\pi}_0 d\theta $ =$\frac{1}{6}(2\pi)$ =$\frac{\pi}{3}$
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