Find the area of the region that lies inside the first curve and outside the second curve.

Question

Find the area of the region that lies inside the first curve and outside the second curve.
r = 9 cos θ, r = 4 + cos θ

 

Please show steps! Thanks in advance!

Answer 1

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.