first off, your limits of integration are wrong. you really need to start looking at the graph of the situation.
let
using symmetry ...
the area shared by the circle r=2 and the cardiod r=2(1-cos(x))
so it looks like the limits of integration are from -pi/2 to pi/2.
squaring both gets 4 and 4-8cos(x)+4cos(x)^2
i'm not sure which is supposed to subtracted from which... but i'll go 4-(4-8cos(x)+4cos(X)^2.
which gives 8cos(x) - 4cos(x)^2
half of that is 4cos(x) -2cos(x)^2
then i can get 4cos(x) - 1 - cos(2x)/2
integrating gives
4sin(x) -x - sin(2x)/4
and plugging in pi/2 and -pi/2 i get 8-pi. but that's not the correct answer that i've been given. where am i going wrong?