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: 22

Answer

The surface is a cone.

Work Step by Step

In cylindrical coordinates, $r$ is always the distance back to the $z$-axis. Note that for a point in the $yz$-plane, $\left| y \right|$ is the distance back to the $z$-axis. So, in the $yz$-plane, $r$ and $y$ behave similarly. Therefore, we can replace $z=r$ with $z=y$ in the $yz$-plane. This is an equation of the line. Since there is no constraint on $\theta$, we graph this line and rotate it around the $z$-axis to obtain the cone as shown in the figure.
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