Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 15 - Section 15.8 - Triple Integrals in Cylindrical Coordinates - 15.8 Exercise - Page 1050: 8


Sphere whose center is at $(0,0,\dfrac{1}{2})$ with radius $\dfrac{1}{2}$

Work Step by Step

Conversion of rectangular to spherical coordinates is as follows: $x=\rho \sin \phi \cos \theta; y=\rho \sin \phi \sin \theta;z=\rho \cos \phi$ and $\rho=\sqrt {x^2+y^2+z^2}$; $\cos \phi =\dfrac{z}{\rho}$; $\cos \theta=\dfrac{x}{\rho \sin \phi}$ Here, $\rho =\cos \phi$ This implies that $\rho^2=\rho \cos \phi$ or, $x^2+y^2+z^2=z$ or, $x^2+y^2+(z-\dfrac{1}{2})^2=\dfrac{1}{4}$ or, $x^2+y^2+(z-\dfrac{1}{2})^2=(\dfrac{1}{2})^2$ This shows that a sphere whose center is at $(0,0,\dfrac{1}{2})$ with radius $\dfrac{1}{2}$
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