[Math] Find the area of the region inside the inner loop of the​ limaçon

Question

Find the area of the region inside the inner loop of the​ limaçon $r=7+14\cos(\theta)$

So doing this problem, I got B the integral from $0$ to $2\pi$: $\dfrac{1}{2} (7+14\cos(\theta))^2$ and the area as $98\pi$. Is the correct way and correct answer? I will also upload my work to get $98\pi$. Thanks for any feedback and help.

Answer 1

Here is a polar plot of the function $r(\theta) =7+14 \cos \theta$ for $\theta \in [0, 2 \pi)$.

Note that the curve passes through the origin when $r(\theta) = 0$. Solving this for $\theta \in [0, 2 \pi)$ gives solutions $\pi \pm { \pi \over 3}$.

Hence you need to compute $A = {1 \over 2}\int_{2 \pi \over 3}^{4 \pi \over 3} r^2(\theta) d \theta$.

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Computing gives $A = {7^2 \over 2} (2 \pi -3 \sqrt{3})$.