Find the area of the region that lies interior to all three circles:

I graphed it, but I have problems figuring out how to solve it. Any help with some explanations will be appreciated!

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- Mar 19th 2008, 08:49 PMhasanbalkan[SOLVED] Area computations in polar coordinates
Find the area of the region that lies interior to all three circles:

I graphed it, but I have problems figuring out how to solve it. Any help with some explanations will be appreciated! - Mar 20th 2008, 10:50 AMroy_zhang
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.

Roy - Mar 21st 2008, 11:47 AMhasanbalkan
Thank you a lot! I was trying to split it into two parts rather than to three parts. Just one correction there in the integrals for people interested in the same problem:

- Mar 21st 2008, 11:53 AMroy_zhang