How to find the area of intersection of three rings/annulus? How to find the area of the intersection/overlap-region of three rings/annuli?
The center of the rings is positioned along a straight line.
The center of the adjacent rings is equally distant.
The width of the rings is equal.
The problem is illustrated in the figure:

The area of the overlap region of two annuli is addressed in Two-Rings. My question adds another annulus/ring to the system. The analogy between two- and three-rings problems is illustrated in Figure: 
The problem breaks down to finding the area of regions labeled as E and F.
 A: Here is a general method for breaking the problem up into simple pieces.
Using Green's Theorem we can convert $$Area(D)=\int \int _D dx dy =\oint_{\partial D} x \ dy$$ and evaluate the latter path integral by parametrizing each part of the oriented boundary $\partial D$ using the usual method for parametrizing circular arcs.  Look at any single oriented circular arc.  In your case, where centers are on the $x$ axis, the parametrization of a counter-clockwise arc looks like   this: $ x= R \cos t + a, y= R\sin t $ where $a$ is the center of the relevant circle and $R$ is its radius.
The arc integral takes the form $\int ( a+ R \cos t ) \cos t \ dt$ which is an elementary integral.
(Incidentally you can check that this arc integral can be interpreted as the (signed) area of a simple region.) The signs are determined by the orientations of the arcs. The last  tricky part is finding the limits of integration in the $t$ variable on each arc, which in your case seems doable with a bit of trigonometry.
