1. By greens theorem, find an expression for the area of the region R, and apply this to evaluate the area of the ellipse bounded by the curve.
x= a cos theta, y=b sin theta
Parametrize the curve as $\displaystyle \bold{g}(\theta) = (g_1(\theta),g_2(\theta))$ for $\displaystyle 0\leq \theta \leq 2\pi$ where $\displaystyle g_1(\theta)=a\cos \theta$ and $\displaystyle g_2(\theta) = b\sin \theta$.
Then, $\displaystyle \iint_R 1 dA=\frac{1}{2}\int_C -ydx + xdy$.
Now evaluate that line integral to get the answer.