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.

Think about it this way. You start off with a circle of radius $a$ of which you know that it has area $\pi \cdot a^2$. Now you pick a direction (say horizontally for concreteness) and stretch the circle in that direction so that what used to be the diameter of length $2a$ will afterwards have length $2b$. Consequently, every line that lies horizontally will have been stretched by a factor of $b/a$, while you leave the vertical direction invariant. Then your total area will also have been changed by a factor of $b/a$, hence yielding $\pi \cdot a \cdot b$. 
