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: 1



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 $div F=\dfrac{\partial A}{\partial x}+\dfrac{\partial B}{\partial y}+\dfrac{\partial C}{\partial z}$ $I=\int_0^1\int_0^1\int_0^1 (3x+3) dz dy dx =\int_0^1\int_0^1 [3xz+3z]_0^1 dy dx =\int_0^1\int_0^1 [3x(1)+3(1)-0] dy dx $ $=\int_0^1\int_0^1 (3x+3) dy dx $ $=\int_0^1 (3x+3)dx $ $=[(\dfrac{3}{2})x^2+3x]_0^1$ $=(\dfrac{3}{2})(1^2-0)+3(1-0)$ Thus, we have $=\dfrac{9}{2}$
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