Answer
$\frac{4 \pi}{3}$
Work Step by Step
Area enclosed by one loop is
$\int_{-\pi/6}^{\pi/6}\frac{r^{2}}{2}d \theta=\int_{-\pi/6}^{-\pi/6}\frac{(4cos(3\theta))^{2}}{2}d \theta$
$=\int_{-\pi/6}^{\pi/6}8cos^{2}(3\theta)d \theta$
$=8\int_{-\pi/6}^{\pi/6}(\frac{(1+cos(6\theta))}{2})d \theta$
$=\frac{4 \pi}{3}$
