Question

Find the area of the region between the loops of the limacon r=5/2 + 5cos(theta).

Answer 1

The area inside the smaller loop is\[A_1 = \int_{2\pi/3}^{4\pi/3}\frac{1}{2}\left(\frac{5}{2}+5\cos\theta\right)^2d\theta = \frac{25}{8}\left(2\pi-3\sqrt{3}\right)\]and the area enclosed by the limaçon is\[A_2 = \int_{-2\pi/3}^{2\pi/3}\frac{1}{2}\left(\frac{5}{2}+5\cos\theta\right)^2d\theta = \frac{25}{8}\left(3\sqrt{3}+4\pi\right)\]so the area between the loops is\[A_2 - A_1 = \frac{25}{4}\left(3\sqrt{3} + \pi\right).\]