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