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: 11



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 $$ Flux =\iiint_{o} -x \space dA \\=\nabla \cdot F \\ =\int_{0}^{2}\int_{0}^{\sqrt {16-4x^2}}\int_{0}^{4-y} (-x) \space dz \space dx \space dy \\\int_{0}^{2}(\dfrac{x\sqrt {16-4x^2}}{2}-4x (16-4x^2)^{1/2} \space dx \\= \dfrac{-40}{3}$$
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