### Home > APCALC > Chapter 11 > Lesson 11.2.1 > Problem11-46

11-46.

What percent of the area inside the limaçon $r = 3 + 6\cos\left(θ\right)$ lies within the inner loop? .

Determine where the graph intersects the pole (origin):

$0=3+6\cos(\theta)$

$\cos(\theta)=-\frac{1}{2}$

$\theta=\pm\frac{2}{3}$

The full area is:

$2\int_0^{\frac{2\pi}{3}}\frac{1}{2}(3+6\cos(\theta))^2d\theta$

The area of the inner loop is:

$2\int_{\frac{2\pi}{3}}^\pi\frac{1}{2}(3+6\cos(\theta))^2d\theta$

Determine the percent of the area that is inside the limacon.