Let C be the positively oriented circle . Use Green's Theorem to evaluate the line integral .
Here, $\displaystyle f\left(x,y\right)=4y$ and $\displaystyle g\left(x,y\right)=11x$
Thus, $\displaystyle \frac{\partial f}{\partial y}=4$ and $\displaystyle \frac{\partial g}{\partial x}=11$
Thus, by Green's Theorem,
$\displaystyle \oint_C 4y\,dx+11x\,dy=7\iint\limits_R\,dA$
Note that the Region R is just a circle of radius 1!!! Thus, $\displaystyle \oint_C 4y\,dx+11x\,dy=7\iint\limits_R\,dA=\dots$
Does this make sense?