Calculus: Early Transcendentals 9th Edition

Published by Cengage Learning
ISBN 10: 1337613924
ISBN 13: 978-1-33761-392-7

Chapter 15 - Section 15.8 - Triple Integrals in Cylindrical Coordinates - 15.8 Exercise - Page 1107: 45

Answer

$\dfrac{4096 \pi}{21}$

Work Step by Step

Apply the spherical coordinates system as: $x=\rho \sin \phi \cos \theta \\ y=\rho \sin \phi \sin \theta \\z=\rho \cos \phi$; So, $\rho=\sqrt {x^2+y^2+z^2} \implies \rho^2=x^2+y^2+z^2$ The jacobian for spherical coordinates is $\phi^2 \sin \phi$ Therefore, $\int_0^{2 \pi} \int_0^{\pi/2} \int_0^{4 \cos \phi} \rho^3 i \rho^2 \rho^2 \sin \phi d\rho d \phi d \theta=\int_0^{2 \pi} \int_0^{\pi/2} [ \rho^6/6]_0^{4 \cos \phi} \sin \phi d\rho d \phi d \ \theta$ Now, set $ \cos \phi =a$ Thus, $E=\dfrac{4^6}{6} \times 2 \pi (\dfrac{-\cos^7 \pi}{7})]_0^{\pi/2} = \dfrac{4096 \pi}{21}$
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