Answer
$$\rho=\tan\phi \sec\phi \cos\theta.$$
Work Step by Step
We apply the equations of this section (page 699).
We have
$$x=z^2 \Longrightarrow \rho\sin\phi\cos\theta=\rho^2\cos^2\phi .$$
That is $$\rho=\tan\phi \sec\phi \cos\theta.$$
Calculus (3rd Edition)
by Rogawski, Jon; Adams, Colin
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: 67
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