## Calculus 8th Edition

Published by Cengage

# Chapter 16 - Vector Calculus - Review - Exercises - Page 1189: 15

#### Answer

Green's Theorem has been verified. $$\int_C Pdx+Qdy=\int\int_D(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dA=0$$

#### Work Step by Step

Green's Theorem states that $$\int_C Pdx+Qdy=\int\int_D(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dA$$ $$\int_C Pdx+Qdy=\int_{C_1}Pdx+Qdy+\int_{C_2}Pdx+Qdy=0$$ and $$\int\int_D(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dA=\int\int_D(-2xy-2xy)=-\int\int_D4xydA$$ $$=\int_{-1}^{1}\int_{x^2}^{1}4xydydx$$ $$\int\int_D(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dA=0$$ Green's Theorem has been verified. $$\int_C Pdx+Qdy=\int\int_D(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dA=0$$

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