Please see the attached graph. We have 3 circles: (in red), (in blue) and (in green). I labeled 2 intersection points of the common interior area of the three circles. They are and or in polar coordinates and
Look closely to the common interior region among the three circles, we know its area can be obtained by adding three areas together: 1) the area obtained by integrating the green circle ( ) from to , 2) the area obtained by integrating the red circle ( ) from to , 3) the area obtained by integrating the blue circle ( ) from to .
Mathematically, we have:
Note that the first and third area are identical, so we only need to calculate one of those.