Once I figured out the solution, I was kicking myself for not figuring it our earlier =)
I think I went wrong in 2 ways. My first attempt was to solve using calculus, and though I came up with a solution for the 2 circle case - I totally failed at finding the integral for 3+ circles.
My bigger mistake was to google for papers after I failed initially. I looked for other peoples venn diagram solutions, and found that they were using approximation techniques to calculate. Since the people writing these papers have a much better math pedigree than I do, it made me think I should use an approximation too.
While I did waste a bit of extra time on this, most of the time was spent in writing javascript code. Since the whole point of the venn diagram library in the first place was to learn javascript, I can’t really say it was a total waste. Also I used the quad tree estimate to verify the exact solution.