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: 66

Answer

$\frac{3}{8}$

Work Step by Step

We calculate the average as: average=$\frac{1}{\frac{2\pi}{3}}\int^{2\pi}_0 \int^{\pi/2}_0 \int^1_0 p^3 cos\phi sin\phi dp $ $ d\phi $ $ d\theta $ =$\frac{3}{8\pi} \int^{2\pi}_0 \int^{\pi/2}_0 cos\phi \sin \phi d\phi $ $ d\theta $ =$\frac{3}{8\pi}\int^{2\pi}_0 [\frac{sin^2\phi}{2}]^{\pi/2}_0d\theta $ =$\frac{3}{16\pi}\int^{2\pi}_0 $ =$(\frac{3}{16\pi})(2\pi)$ =$\frac{3}{8}$
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