Answer
$\dfrac{9 \pi}{2}$
Work Step by Step
Since, $L=\int_{0}^{3\pi/2}\sqrt{(\dfrac{dx}{d\theta})^2+(\dfrac{dy}{d\theta})^2}d\theta$
Thus, $L=\int_{0}^{3\pi/2} \sqrt{3(\sin^2 \theta+\cos^2 \theta)} d\theta$
Then, we have $L=\int_{0}^{3\pi/2} \sqrt{3} d\theta$
Thus, $L =3(\dfrac{3\pi}{2}-0)=\dfrac{9 \pi}{2}$
