Let C be the positively oriented square with vertices , , , . Use Green's Theorem to evaluate the line integral .
Note that $\displaystyle f\left(x,y\right)=7y^2x$ and $\displaystyle f\left(x,y\right)=6x^2y$
Thus, $\displaystyle \frac{\partial f}{\partial y}=14xy$ and $\displaystyle \frac{\partial g}{\partial x}=12xy$
Also note that the region R is bounded by the lines $\displaystyle x=0,~x=2,~y=0,~y=2$
Thus, by Green's Theorem,
$\displaystyle \oint_C7y^2x\,dx+6x^2y\,dy=-2\int_0^2\int_0^2xy\,dx\,dy$
Can you take it from here?