University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 14 - Section 14.7 - Triple Integrals in Cylindrical and Spherical Coordinates - Exercises - Page 805: 60


$$32\pi $$

Work Step by Step

The paraboloid will intersect the $ xy $-plane when $9-x^2-y^2=0 \implies x^2+y^2=9$ Our aim is to integrate the integral as follows: $$ volume=(4) \times \int^{\pi/2}_0 \int^3_1 \int^{9-r^2}_0 dz \space r dr \space d\theta \\=(4) \times \int^{\pi/2}_0 \int^3_1 (9r-r^3)dr \space d\theta \\=(4) \times \int^{\pi/2}_0 [\dfrac{9}{2} \times r^2-\dfrac{r^4}{4}]^{3}_1 d\theta \\=(4) \int^{\pi/2}_0 (\dfrac{81}{4}-\dfrac{17}{4})d\theta\\=32\pi $$
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