Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 13 - Vector Geometry - 13.7 Cylindrical and Spherical Coordinates - Exercises - Page 700: 49


$\rho\lt 1$.

Work Step by Step

The set $x^2+y^2+z^2\leq1$ is the closed sphere in $R^3$, centered at the origin. Since $$ \begin{aligned} &x=\rho \sin \phi \cos \theta\\ &y=\rho \sin \phi \sin \theta\\ &z=\rho \cos \phi, \end{aligned} $$we have $$x^2+y^2+z^2=\rho^2(\sin^2\phi\cos^2\theta+\sin^2\phi\sin^2\theta+\cos^2\phi)\\ =\rho^2$$ That is, $\rho^2\leq 1$ and since $\rho$ is non negative then $\rho\lt 1$.
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