<s>[INST] Area inside outer curve of limacon $r=1+2\cos\theta$ I need to find the area inside the outer curve of limacon $r=1+2\cos\theta$
Here is what I tried:
$$
0=1+2\cos\theta
$$
$$
\theta=\cos^{-1}(-\frac{1}{2})
$$
$$
\theta=\frac{2}{3}\pi  ,  \theta=\frac{8}{3}\pi
$$
$$
A = 2\int^{\frac{8}{2}\pi}_{\frac{2}{3}\pi}r^2d\theta=\frac{r^3}{3}|^{\frac{8}{3}\pi}_{\frac{2}{3}\pi}=2\frac{(1+2\cos\theta)^3}{3}|^{\frac{8}{3}\pi}_{\frac{2}{3}\pi}=0
$$
so, 0 area is clearly wrong. Where was my mistake?
 [/INST] $\int\limits_{\theta = 0}^{2 \pi} (1 + 2 \cos (\theta )\ d \theta = 2 \pi$.
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