Calculate area of Ellipse without calculus? I like the way integration works, but the final formula $\pi ab$ is too simple.
I know there is a more deeper way to derive it. I just don't like to use calculus here, too many equations. 
I'd like to use simple math, which does offer deeper insight into it.

You may use an affine map $\varphi$ to send an ellipse into a circle. Since affine maps preserve the ratios between areas, the area of the ellipse is $\frac{\text{Area}(\text{circle})}{\left|\det\varphi\right|}=\pi a b$.
