Using polar coordinates to find the area of an ellipse Considering an ellipse with the $x$ radius equal to $a$ and the $y$ radius equal to b$:$
I figured that some kind of parameterization might be: 
$x=a\cos\theta$
$y=b\sin\theta$
and then polar $r^2$ is just $x^2 + y^2$
But then I tried to come up with some unit of infinitesimal area using triangles $\left(\dfrac{d\theta r^2}{2}\right)$ which does not give the correct answer. I read somewhere that my polar coordinates are wrong and that they are actually 
$x=ar\cos\theta$
$y=br\sin\theta$
But this does not make sense to me as an engineer because that seems like it would have the dimension of area equal to the dimension of a distance. The integral also takes $r$ from $0$ to $1$ which I thought was eliminated because the equation for $r$ should be in terms of $\theta$ and the constants $a$ and $b.$ 
I would like some explanation of what I am doing wrong that would make some "physical" sense (or why physical intuition might fail for this problem)

My answer is linked here.

My name is Marco mamello10@gmail.com. 
