Let $\bar F(x^2-xy)i+(x^2-y^2)j$ be the function and C be the boundary of region bounded by

$x^2=2y\ \ \ and\ \ x=y$

$p=x^2-xy$

$Q=x^2-y^2$

$\dfrac{\partial p}{\partial y}=-x$

$\dfrac{\partial Q}{\partial x}=2x$

By Green Theorem, 

$\int_c\ (p\ dx+Q\ dy)=\int \int _R\Bigg(\dfrac{\partial Q}{\partial x}-\dfrac{\partial p}{\partial x}\Bigg)dx\ dy$..............................(1)

Now consider 0 to 2;     $y=\dfrac{x^2}{2}\ \ to\ \ x$

We get  $\int \int _R\Bigg(\dfrac{\partial Q}{\partial x}-\dfrac{\partial p}{\partial y}\Bigg)dx\ dy=\int_0^2 \int_{x^2/2}^x(2x+x)dxdy=3\int_0^{2}\int_{x^2/2}^xxdy\ dx =3\int_0^2x[y]_{x^2/2}^x\ dx$

                                                        $=3\int _0^2x\Bigg[x-\dfrac{x^2}{2}\Bigg]dx=3\int \Bigg[x-\dfrac{x^2}{2}\Bigg]dx=3\Bigg[\dfrac{x^3}{3}-\dfrac{x^4}{8}\Bigg]_0^2=\Bigg[\dfrac{8}{3}-\dfrac{16}{8}\Bigg]=2$

$\int\int_R\Bigg(\dfrac{\partial Q}{\partial x}-\dfrac{\partial p}{\partial y}\Bigg)dx\ dy=2$.........................................(2)

Consider along OA 

$x^2=2y$

$\therefore 2x\ dx=2\ dy$

$x\ dx= dy$

$\therefore x=0\ to\ 2$

We have, 

$\int_{OBA}(Pdx+Qdy)=\int_0^2\Bigg[x^2-x\Bigg(\dfrac{x^2}{2}\Bigg)\Bigg]dx+\Bigg[x^2-\Bigg(\dfrac{x^2}{2}\Bigg)^2\Bigg]x\ dx$

                            $=\int_0^2\Bigg[x^2-\dfrac{x^3}{2}+x^3-\dfrac{x^5}{4}\Bigg]dx=\int_0^2\Bigg[x^2+\dfrac{x^3}{2}-\dfrac{x^5}{4}\Bigg]dx$

                            $=\bigg[\dfrac{x^3}{3}+\dfrac{x^4}{8}-\dfrac{x^6}{24}\Bigg]_0^2$

                            $=\dfrac{8}{3}+\dfrac{16}{8}-\dfrac{64}{24}=\dfrac{64+48-64}{24}=\dfrac{48}{24}=2$

$\therefore \int_{OBA}(Pdx+Q dx)=2$......................(3)

Along OA,    x = y

$\therefore \ dx=dy\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x = 2\ to \ 0$

$\int_{AO}\ (Pdx+Qdy)=0$          ........................(4)

$\therefore \int_C(Pdx+Qdy)=\int_{OBA}(Pdx+Qdy)+\int_{AO}(Pdx+Q\ dy)=2+0=2$

$\int_C(Pdx+Qdy)=2$   ........................(5)

From (3) and (5) $\int_C(Pdx+Q\ dy)=\int\int _R\Bigg(\dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y}\Bigg)dx\ dy$

Hence Green's theorem is verified.