Answer
Total distance is $\,6{\sqrt{2}}$
Work Step by Step
$\begin{aligned} & \frac{d x}{d t}=2 \sin (t) \cos (t)=\sin (2 t) \\ & \frac{d y}{d t}=-2 \sin (t) \cos (t)=-\sin (2 t) \\ & L=\int_0^{\pi / 2} \sqrt{\left(\frac{d x}{d t}\right)^2+\left(\frac{d y}{d t}\right)^2} d t \\ & L=\int_0^{\pi / 2} \sqrt{\sin ^2 2 t+\sin ^2 2 t} d t \\ & L=\int_0^{\pi / 2} \sqrt{2 \sin ^2 2 t} d t \\ & \text { is } L=\sqrt{2} \int_0^{\pi / 2}(\sin 2 t) d t \\ & L=\sqrt{2}\left[-\frac{\cos 2 t}{2}\right]_0^{\pi / 2} \\ & L=\sqrt{2}\left[-\frac{\cos \pi}{2}+\frac{\cos 0}{2}\right] \\ & L=\sqrt{2}\left[\frac{1}{2}+\frac{1}{2}\right]=\sqrt{2} \\ & \text { total distance }=6 L=6 \sqrt{2} \\ & \end{aligned}$
