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 804: 28



Work Step by Step

Our aim is to integrate the integral as follows: $$\int^{\pi/3}_{\pi/6} \int^{2 \csc\phi}_{\csc\phi} \int^{2\pi}_0 p^2\sin \phi \space d\theta \space dp \space d\phi =2\pi \int^{\pi/3}_{\pi/6} \int^{2 \csc\phi}_{\csc\phi} p^2 \space\sin\phi dp \space d\phi \\=\dfrac{2\pi}{3}\int^{\pi/3}_{\pi/6} [ p^3 \times \sin\phi]^{2 \csc\phi}_{\csc\phi} \space dp \\=\dfrac{14\pi}{3} \times \int^{\pi/3}_{\pi/6} \csc^2\phi \space d\phi \\=\dfrac{28\pi}{3\sqrt{3}}$$
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