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: 53

Answer

The description of the given set in spherical coordinates is $0 \le \rho \le 2$, ${\ \ }$ $\theta = \frac{\pi }{2}$ or $\theta = \frac{{3\pi }}{2}$.

Work Step by Step

In spherical coordinates: 1. the radial coordinate $\rho$ Since $x=0$ and ${\rho ^2} = {x^2} + {y^2} + {z^2}$, so we have ${\rho ^2} \le 4$. Since $\rho$ is always positive, we have $0 \le \rho \le 2$. 2. the angular coordinate $\theta$ satisfies $\cos \theta = \frac{x}{r} = \frac{x}{{\sqrt {{x^2} + {y^2}} }}$ Since $x=0$, we have $\cos \theta = 0$. So, $\theta = \frac{\pi }{2},\frac{{3\pi }}{2}$ 3. the angular coordinate $\phi$ There is no constraint on $\phi$. Therefore, the description of the given set in spherical coordinates is $0 \le \rho \le 2$, ${\ \ }$ $\theta = \frac{\pi }{2}$ or $\theta = \frac{{3\pi }}{2}$.
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