Calculus 8th Edition

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

Chapter 16 - Vector Calculus - 16.9 The Divergence Theorem - 16.9 Exercises - Page 1186: 14


$4 \pi R^5$

Work Step by Step

Here, $div F=(x^2+y^2+z^2) \cdot \lt x,y,z \gt=3(x^2+y^2+z^2)+2(x^2+y^2+z^2)$ or, $div F=5(x^2+y^2+z^2)=5\rho^2 $ $Flux=\int_{0}^{2 \pi}\int_0^{\pi} \int_{0}^{R} 5\rho^2 dv=\int_{0}^{2 \pi}\int_0^{\pi} \int_{0}^{R} 5\rho^2 \times \sin \phi \times d \rho d\phi d \theta$ $=\int_{0}^{2 \pi}\int_0^{\pi} \int_{0}^{R} 5\rho^2 \times \sin \phi \times d \rho d\phi d \theta$ $=\int_{0}^{2 \pi}\int_0^{\pi} R^5 \times \sin \phi d \rho d\phi d \theta$ $=\int_{0}^{2 \pi} R^5 [(-\cos \phi)_0^{\pi} d \theta$ $=\int_{0}^{2 \pi} R^5 [-(\cos \pi- \cos 0] d \theta$ $Flux=4 \pi (R^5)$
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