Calculus 8th Edition

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

Chapter 16 - Vector Calculus - 16.7 Surface Integrals - 16.7 Exercises - Page 1173: 11


$\dfrac{\sqrt {21}}{3}$

Work Step by Step

When the surface is orientable then the flux through a surface is defined as: $\iint_S F \cdot dS=\iint_S F \cdot n dS$ where, $n$ is the unit vector. Consider $\iint_S x dS =\iiint_{D}x \sqrt {(-4)^2+(2)^2+1} dA$ or, $\iint_S x dS=\sqrt{21} \iint_D dA= \int_{0}^{1} \int_{2x-2}^0 x dy dx \times \sqrt{21}= \int_{0}^{1} [xy]_{2x-2}^0 x dx \times \sqrt{21}$ $\iint_S x dS =- \int_{0}^{1} 2x^2-2x dx \times \sqrt{21}=- \sqrt{21} [(\dfrac{2x^3}{3})-x^2]_{0}^{1}=-\sqrt {21} \times \dfrac{2(1)^3}{3}-(1)^2]_0^1=\dfrac{\sqrt {21}}{3}$
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