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.

Consider the unit disk (bounded by the circle of radius $1$, centered at the origin).  Now, to construct an ellipse whose axes are $a$ along the $x$-axis and $b$-along the $y$-axis.  This corresponds to the application of the linear transformation
$$
\begin{bmatrix}a&0\\0&b\end{bmatrix}.
$$
We can confirm that this is an ellipse because if your original coordinates are $x_1$ and $x_2$ while your new coordinates are $y_1$ and $y_2$, we have $y_1=ax_1$ and $y_2=bx_2$.  Therefore, $y_1$ and $y_2$ satisfy:
$$
\frac{y_1^2}{a^2}+\frac{y_2^2}{b^2}=1.
$$
Since linear transformations scale areas by the determinant (and the original disk has area $\pi$), the resulting area is $ab\pi$.
