You might want to try following through on GEdgar's suggestion. First, verify that
$r=\frac{b^2}{a-\sqrt{a^2-b^2}\cos\,t}$
is the polar equation of an ellipse with semiaxes $a$, and $b$, with the origin as one of the foci. You can then use the formula
$\int_0^{2\pi}\frac{r^2}{2}\mathrm d\theta$
(which is what you'd obtain if you set up the double integral and evaluate before substituting in $r$). In particular, you can simplify things by noting that for this case,
$\int_0^{2\pi}\frac{r^2}{2}\mathrm d\theta=\int_0^\pi r^2\mathrm d\theta$
(why?)