Answer
$A$ = $\pi{ab}$
Work Step by Step
By symmetry of the ellipse about the $x$ and $y$ axes
$A$ = $4\int_a^0{ydx}$ = $-4\int_0 ^{\frac{\pi}{2}}b\sin\theta (-a\sin\theta)d\theta$ = $4ab\int_0 ^{\frac{\pi}{2}}\sin^2{\theta d\theta}$ = $2ab\int_0 ^{\frac{\pi}{2}}(1-\cos2\theta)d\theta$ = $2ab\left[\theta-\frac{\sin2\theta}{2}\right]_0^{\frac{\pi}{2}}$ = $\pi{ab}$
