Ira S. answered • 10/21/16

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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 ∫ R

^{2}- r^{2}dθA = 1/2 ∫ 81cos

^{2}θ - (16+8cosθ +cos^{2}θ) dθ 0 to 30A = 1/2 ∫80 cos

^{2}θ - 8cosθ -16 dθA = ∫ 40 cos

^{2}θ - ∫4cosθ - ∫8 dθ 0 to 30I 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.

Luis O.

02/02/18