Thomas' Calculus 13th Edition

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

Chapter 16: Integrals and Vector Fields - Section 16.8 - The Divergence Theorem and a Unified Theory - Exercises 16.8 - Page 1026: 12


$$\dfrac{12 \pi a^5}{5}$$

Work Step by Step

We know that $$ div F=\dfrac{\partial P}{\partial x}i+\dfrac{\partial Q}{\partial y}j $$ From the given equation, we have $$ Flux =\iiint_{o} 3x^2+3y^2+3z^2 \space dA\\=\nabla \cdot F \\=3 \times \int_{0}^{2 \pi}\int_{0}^{\pi}\int_{0}^{a} (\rho^2) \times (\rho^2 \sin \phi) \space d\rho \space d \phi \space d\theta \\=(3) \int_{0}^{2 \pi}(\dfrac{2a^5}{5} ) \space d\theta \\= \dfrac{12 \pi a^5}{5}$$
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