Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 15: Multiple Integrals - Section 15.7 - Triple Integrals in Cylindrical and Spherical Coordinates - Exercises 15.7 - Page 918: 9

Answer

$\frac{\pi}{3}$

Work Step by Step

$\int^1_0 \int^{\sqrt{z}}_0 \int^{2\pi}_0 (r^2cos^2\theta+z^2)rd\theta $ dr dz =$\int^1_0 \int^{\sqrt{z}}_0 [\frac{r^2\theta}{2}+\frac{r^2 sin2\theta}{4}+z^2 \theta]^{2\pi}_0 rdr dz $ =$\int^1_0 \int^{\sqrt{z}}_0(\pi r^3+2\pi rz^2)dr $ dz = $\int^1_0 [\frac{\pi r^4}{4}+\pi r^2 z^2]^{\sqrt{z}}_0$ dz =$\int^1_0 (\frac{\pi z^2}{4}+\pi z^3)$ dz =$[\frac{\pi z^3}{12}+\frac{\pi z^4}{4}]^1_0$ =$\frac{\pi}{3}$
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