Question on Green's Theorem Consider the vector field $\textbf{f}(x,y)=(ye^{xy}+y^2\sqrt{x})\textbf{i}+(xe^{xy}+\frac{4}{3}yx^{\frac{3}{2}})\textbf{j}$.
Use Green's Theorem to evaluate $\int_C\textbf{f} \dot d\textbf{r}$, where $C$ is the ellipse given by $(x-1)^2+\frac{y^2}{9}=1$, oriented counterclockwise. 
$\int_C\textbf{f} \dot d\textbf{r} = \int \int_R (\frac{\partial Q}{\partial x} -\frac{\partial P}{\partial y}) \ dx \ dy$
$=\int \int_R (xe^{xy}+ye^{xy}+2yx^\frac{1}{2})-(ye^{xy}+xe^{xy}+2\sqrt{x}y) \ dx \ dy$
$=\int \int_R 2y\sqrt{x}-2\sqrt{x}y \ dx \ dy$
What would the bounds be on the integral? I think I should parametrize to polar coordinates, but I'm not sure how to do that when the ellipse's center isn't the origin. Or would I just use $0\leq x \leq 2$ and $-\sqrt{9(1-(x-1)^2)} \leq y \leq  +\sqrt{9(1-(x-1)^2)}$ ? 
Thanks. 

I would highly recommend you do not transform to polar coordinates, you would not get nice arguments for the function you are integrating. 
The bounds $-\sqrt{9(1-(x-1)^2)} \leq y \leq  +\sqrt{9(1-(x-1)^2)}$ and $0\leq x \leq  2$ will work perfectly fine for your integral, you are on the right track the rest is plug and chug.
Do you have any follow up questions on how to proceed from here?
Edit
In light of the recent updates to the question it looks like the function you are integrating is zero, so the boundaries of integration (being finite) do not matter.
