Area between two closed curves

[SSGD]

I have been trying to find an answer to this problem for some time. So I was hoping the community might be able to point me in the right direct.

https://drive.google.com/open?id=1Y2hYkRG94whLroK20Zmce07jslyRL3zZ

I couldn't get the image to load. So above is a link to an image of the problem. I am trying to find a general solution to the intersection of the two general closed loops.

In the image Ai and Aj are just the area of each closed loops and Aij is the area of the intersection of the two closed loops. ri and rj are just parameterizations of the positions around the closed loop.

Their is a general equation for the area of a closed loop defined by a line integral and I was wondering if there might be a line integral for the area of intersection between two closed loops.

 

[mfb]

You can find the boundary of the overlap area by finding the intersections between the two boundaries and combining the sections between them to form a new line.
If there is an infinite set of intersections or similar things get more complicated.