Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 15 - Multiple Integrals - 15.7 Triple-Integrals in Cylindrical Coordinates - 15.7 Exercises - Page 1084: 30



Work Step by Step

$z=9-x^2-y^2$ and $z=9-r^2$ For cylindrical coordinate system: $I=\int_0^{\pi} \int_{0}^{3}\int_{0}^{9-r^2} (r) \times (r)\times dz dr d\theta$ $=\int_0^{\pi} \int_{0}^{3} [r^2z]_{0}^{9-r^2} \times dr d\theta$ $=\int_0^{\pi} \int_{0}^{3} r^2(9-r^2) \times dr d\theta$ $=\int_0^{\pi} \int_{0}^{3} (9r^2-r^4)dr d\theta$ $=\int_0^{\pi} [3r^3-\dfrac{r^5}{5}]_{0}^{3} d\theta$ $=[\dfrac{162\theta}{5}]_0^{\pi}$ $=\dfrac{162\pi}{5}$
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