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 1185: 4


$36 \pi$

Work Step by Step

In order to verify the Divergence Theorem which is true for for the vector field over the region $E$, we will have to add all the surface integrals and should be make sure that all are equal to such as: $\iint_S \overrightarrow{F}\cdot d\overrightarrow{S}=\iiint_Ediv \overrightarrow{F}dV $ Here, we have $S$ shows a closed surface. The region $E$ is inside that surface. We have $div F=\dfrac{\partial A}{\partial x}+\dfrac{\partial B}{\partial y}+\dfrac{\partial C}{\partial z}$ and $div F=\dfrac{\partial (x^2)}{\partial x}+\dfrac{\partial (-y)}{\partial y}+\dfrac{\partial (z)}{\partial z}=2x-1+1=2x$ Consider $I=\iint_{y^2+z^2 \leq 9} \int_0^2 2xdxdydz=\iint_{y^2+z^2 \leq 9} [x^2]_0^2 dydz$ Now, we have $I=\iint_{y^2+z^2 \leq 9} (4) dydz=4 \cdot \iint_{y^2+z^2 \leq 9} dydz=4 \cdot 9=36 \pi$
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.