Area of an ellipse and similar problems

[Gauss M.D.]

To find the area, you break the ellipse into infinitesimaly small triangles and integrate.

But why? Why not break it up into infinitesimaly small circle segments and calculate it through circumference instead?

There are other problems regarding integration of geometric objects that has me wondering the same thing. Integrating a function in polar coordinates for example. It seems to me that the triangle method and the circle method has equal logical merit. But one of them produces the wrong result. What's governing which one to choose?

 

[0xDEADBEEF]

Could you put your circle segment idea in mathematical terms? I am sure that it works if you do it right.