Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 15 - Multiple Integrals - 15.8 Triple Integrals in Spherical Coordinates - 15.8 Exercises - Page 1090: 19


$ \int_{0}^{\pi/2}\int_{0}^{3} \int_0^{2}f(r \cos \theta,r \sin \theta,z) r dz dr d\theta$

Work Step by Step

The conversion of rectangular coordinates to spherical coordinates is given as: $x=\rho \sin \phi \cos \theta; y=\rho \sin \phi \sin \theta;z=\rho \cos \phi$ Here, $\rho=\sqrt {x^2+y^2+z^2}$; $\phi =\cos^{-1} [\dfrac{z}{\rho}]; \theta=\cos^{-1}[\dfrac{x}{\rho \sin \phi}]$ The conversion of rectangular to cylindrical coordinate system is : $r^2=x^2+y^2 \\ \tan \theta=\dfrac{y}{x} \\z=z$ Now, $x=r \cos \theta; y=r \sin \theta, z=z$ and $\iiint f(x,y,z) dz r dr d\theta=\iiint f(r \cos \theta,r \sin \theta,z) dz r dr d\theta$ Need to plug the boundaries, then we get $\iiint f(r \cos \theta,r \sin \theta,z) dz r dr d\theta= \int_{0}^{\pi/2}\int_{0}^{3} \int_0^{2}f(r \cos \theta,r \sin \theta,z) r dz dr d\theta$
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