--ChrisOk, so I have tried this problem like 4 times and have not gotten the correct answer yet. The instructions say, "Use a double integral to find the area of the shaded region."
Then there is a graph with the entire limacon filled in.
This is what I have done but I haven't gotten the correct answer.
∫ (1/2 r^2 |(0 to 2(1-cosθ)) )dθ
∫ (1/2 *4 (sin^2(θ)))dθ This is where your mistake is: !!!
∫ (2 sin^2(θ))dθ
∫ (2*-1/2 (cos(2θ)-1))dθ
∫ (-cos(2θ) + 1)dθ
-1/2 ∫ (cosu)du + ∫1dθ
-1/2 (sin2θ) | (0 to 2π) + θ | (0 to 2π)
0 - 0 + 2π - 0 = 2π
Why is this not correct? It is supposed to be 6π. Where have I made a mistake?