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: 27

Answer

$2\pi $

Work Step by Step

$\int^2_0 \int^0_{-\pi} \int^{\pi/2}_{\pi/4} p^3 \sin2\phi d\phi d\theta $ $ dp $ =$\int^2_0 \int^0_{-\pi}p^3[\frac{-cos2\phi}{2}]^{\pi/2}_{\pi/4} d\theta dp $ =$\int^2_0 \int^0_{-\pi} \frac{p^3}{2}d\theta dp $ =$\int^2_0 \frac{p^2\pi}{2}dp $ =$[\frac{\pi p^4}{8}]^2_0$ =$2\pi $
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