Find the area of the region that consists of all points that lie within the circle r = 1 but outside the polar equation r = cos(2θ)
A = ∫ 1/2 (r2^2 - r1^2) dθ, where r2 is outer curve and r1 is inner curve.
The Attempt at a Solution
Here is what the graph looks like if you want to see it: http://www.wolframalpha.com/input/?i=r+=+cos(2x)+polar
Ok so first I had to find out the limits for my integral. I set: 1 = cos(2θ) and got that θ = ∏.
Now I made my integral, but instead of going from -∏ to ∏ I went from 0 to ∏ and multiplied it by 2
A = 2 ∫ 1/2 [(1^2) - (cos(2θ)^2)] dθ, from 0 to ∏
A = ∫ [ (1) - (1/2 cos(4θ) +1)] dθ, from 0 to ∏
I solved it and received: ∏ - ∏/2
Which equals ∏/2.
Just want to know if I did it right. The main part I was worried about was the limits for my integral.