I have to find the area between a line and a cardioid, given by $\rho_1 = 8 + 8 \sin(\theta)$ and $\rho_2 = 4/\hspace{-0.5mm}\sin(\theta)$.
First I found when both are positive, which is always for the first one, and only from $0$ to $2\pi$ for the second one. Then I graph them both and now I have to find the intersections, but I get $8+\sin(\theta) = 4/\hspace{-0.5mm}\sin(θ)$, which is $\sin^2(\theta) + 1 = 1/2$, and I have no idea how to solve that.
Then, for the area, I know it's the top part of the graphic, so let's say they intersect in $\theta=a$ and $\theta=b$, so I have $$\int_{a}^{b} (8+8\sin\theta)^2 - (4/\hspace{-0.5mm}\sin\theta)^2 d\theta$$ for the cardioid is over the line. Is that correct? I'm not supposed to solve it, just to propose it.
And that's all, I hope you can help me. Thank you!