Multivariable Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 0-53849-787-4
ISBN 13: 978-0-53849-787-9

Chapter 15 - Multiple Integrals - 15.7 Exercises - Page 1050: 45

Answer

$\dfrac{1}{2}\pi kha^4$

Work Step by Step

Here, $I_z=\iiint_{E} (x^2+y^2) \rho(x,y,z) dV=k\int_{0}^{2 \pi} \int_{0}^{h}\int_{0}^{a}(r^2)[r dr ] dz d\theta $ $=k\int_{0}^{2 \pi} \int_{0}^{h}\int_{0}^{a}r^3 dr dz d\theta $ $=k \int_{0}^{2 \pi} d\theta \times k\int_{0}^{h} dz \times k \int_{0}^{a}(r^3) dr $ $=(2\pi k) [z]_0^h [\dfrac{r^4}{4}]_0^a $ $=2\pi k (h-0) [\dfrac{a^4}{4}-0] $ $=2\pi k (h) (\dfrac{a^4}{4}) $ Hence, $I_z=\dfrac{\pi kha^4}{2}$
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