I've run into a geometry problem that feels like it should have an easy answer. But short of numerical integration, I can't find a way to solve it.
Consider a filled circle on top of a filled ellipse with their origins overlapping, and the major axis of the ellipse on the X axis (minor axis aligned with y). (See picture)
The radius of the circle and major/minor axes of the ellipse are known constants. Find the green shaded area.
If at all possible, I would like a function $A(Green) = f(a,b,R)$. I've tried finding the intersection point of the two shapes in the first quadrant and integrating, but the integrand does not lend itself to a simple analytical integration technique, and Mathematica has been chewing on this problem for about an hour now to no success.
Numerical integration could work for my purposes, and I may need to resort to that. Is there a better way to proceed before I do that?