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: 24

Answer

Please see the figure attached.

Work Step by Step

In cylindrical coordinates, $r$ is always the distance back to the $z$-axis. Thus, $1 \le r \le 3$ corresponds to the region outside of a circle of radius $1$ and inside of a circle of radius $3$. Since the $z$-coordinate is the same as in rectangular coordinates, $0 \le z \le 4$ corresponds to region between the plane at height $z=0$ and the plane at height $z=4$. So, it is a cylinder. However, since $\theta$ is constrained in the interval $0 \le \theta \le \frac{\pi }{2}$, the region is located in the first quadrant.
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