University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 14 - Section 14.7 - Triple Integrals in Cylindrical and Spherical Coordinates - Exercises - Page 805: 66



Work Step by Step

Our aim is to integrate the integral as follows: $$ Average= \dfrac{1}{(2 \pi/3)} \times \int^{2\pi}_0 \int^{\pi/2}_0 \int^1_0 p^3 \cos(\phi) \space \sin(\phi) \space dp \space d\phi \space d\theta \\=\dfrac{3}{8\pi} \times \int^{2\pi}_0 \int^{\pi/2}_0 \cos(\phi) \sin( \phi) \space d\phi \space d\theta \\=\dfrac{3}{8\pi} \times \int^{2\pi}_0 [\dfrac{sin^2\phi}{2}]^{\pi/2}_0 (1) d\theta \\=\dfrac{3}{16\pi} \times \int^{2\pi}_0 (1) d\theta \\=(\dfrac{3}{16\pi})(2\pi)\\=\dfrac{3}{8}$$
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