R=2 and r=3+2cos(theta)
What have you tried?
Here's a hint:
1) Draw both polar equations on the same graph from 0 to 2*pi.
2) Find the interval for which R=2 > R=3+2cos(theta), and the interval for which R=3+2cos(theta) > R=2.
3) Try shading the area that is enclosed by both curves, and finding integrals which represent these shaded areas on their intervals.
You should see that the area you have shaded is actually the area under the lesser of the two curves from point A to point B.
Thus, you need to find the area under that shaded lesser curve on each interval.
Recall that the area under a polar curve is given by integrating (1/2)(r^2) from theta=A to theta=B
In this case, the endpoints will be where the two curves intersect on the graph, as you should see that the lesser curve switches after these intersections. Calculate the integral for each shaded section separately, then add them together.