Calculus 8th Edition

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

Chapter 16 - Vector Calculus - 16.4 Green's Theorem - 16.4 Exercises - Page 1142: 7

Answer

$\dfrac{1}{3}$

Work Step by Step

Green's Theorem states that: $\oint_C M \,dx+ N \,dy=\iint_{D}(\dfrac{\partial N}{\partial x}-\dfrac{\partial M}{\partial y})dA$ The integrand of the double integral becomes: $\oint_C (y+e^{\sqrt {x}} ) dx+(2x+\cos y^2) dy=\iint_{D}(\dfrac{\partial (2x+\cos y^2)}{\partial x}-\dfrac{\partial (y+e^{\sqrt {x}} ) }{\partial y}) \ dA$ $=\int_{0}^{1} \int_{x^2}^{\sqrt x} (2-1) \ dy \ dx$ $=\int_{0}^{1} [y]_{x^2}^{\sqrt x} \ dx$ $=\int_0^1 [\sqrt x-x^2 ] \ dx$ $=\dfrac{2}{3} [ x^{(3/2)}]_0^1-\dfrac{1}{3}[x^3]_0^1$ $=\dfrac{1}{3}$
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