I hope you have a graphing calculator so that you can see what these graphs look like. More importantly, sometimes polar equations trace over already graphed.
So r=9cosΘ is a circle with a radius 9
and r = 4+ cosθ is a cardioid.
Just like in the xy plane, you need to find the point of intersections.
9cosθ = 4 + cosθ
cos θ = .5
θ = 30 and 330.
I would just use the limits of integration from o to 30 and then double it base on symmetry. (or -30 to 30).
A=1/2 ∫ R2 - r2 dθ
A = 1/2 ∫ 81cos2θ - (16+8cosθ +cos2θ) dθ 0 to 30
A = 1/2 ∫80 cos2θ - 8cosθ -16 dθ
A = ∫ 40 cos2θ - ∫4cosθ - ∫8 dθ 0 to 30
I hope that you can do the rest because i"m tired of typing math. The first integral is integration by parts...and the rest is straight foward. Plug in 30...the 0 and the don't forget to double it due to symmetry.
Hope this helped.