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.
$$\int_C Pdx+Qdy=\int\int_D(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dA=0$$
