allaboutmusic

I'm not going to be much help - I'd have done it the same way described.

I would find the distance between the two intersection points, get the area of the two sectors (defined by the radii of the two circles and the arcs between the two intersection points), and subtract the area of the two triangles (defined by the radii of the two circles and the chord between the two intersection points).

I have a feeling some optimisation can be done based on finding the length of the chord, and using basic trigonometry to simplify the formula to find the area.