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: 62

Answer

Please see the figure attached.

Work Step by Step

In spherical coordinates, $\rho$ is the distance back to the origin. Thus, $\rho=1$ describes a sphere of radius $1$. While $\phi$ is the angle of declination, which measures how much the ray through a point $P$ on the sphere declines from the vertical. Since it is in the interval $\frac{\pi }{3} \le \phi \le \frac{{2\pi }}{3}$, rotating it around the $z$-axis we obtain the part of the sphere as is shown in the figure.
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