Show that the area of ellipse
x^2/a^2 + y^2/b^2 =1 is pi*ab.
It's symmetric with respect to the $\displaystyle y$ axis.
We consider to compute the half area of the ellipse, then when you get the final answer, multiply by 2 and you are done.
It remains to compute $\displaystyle \int_{-a}^a\sqrt{b^2\left(1-\frac{x^2}{a^2}\right)}\,dx$