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: 23



Work Step by Step

Since, we have $\iint_S F \cdot dS=\iint_D [P \dfrac{\partial h}{\partial x}-Q+R\dfrac{\partial h}{\partial z}]dA $ where, $D$ is projection of $S$ onto xz-plane. $\iint_S F \cdot n dS=\iint_D -xy (-2x) -(yz) (-2y) +(zx) dA= \int_{0}^{1} \int_0^{1} [2x^2y+2y^2z+zx] dy dx$ and $\iint_S F \cdot n dS= \int_{0}^{1} \int_0^{1} 2x^2y+2y^2 \times (4-x^2-y^2)+(4-x^2-y^2) \times (x) dy dx$ or, $\iint_S F \cdot n dS= \int_{0}^{1} x^2+(\dfrac{8}{3}) -(\dfrac{2x^2}{3})-(\dfrac{2}{5}) +4x-x^3-\dfrac{x}{3} dx$ or, $\iint_S F \cdot n dS=[\dfrac{34x}{15}+\dfrac{x^3}{9}+\dfrac{11x^2}{6}-\dfrac{x^4}{4}]_{0}^{1}=\dfrac{713}{180}$
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