Answer
$L = 12$
Work Step by Step
$x$ = $3\cos t-\cos3t$
$y$ = $3\sin t-\sin 3t$
$\frac{dx}{dt}$ = $-3\sin t +3\sin 3t$
$\frac{dy}{dt}$ = $3\cos t-3\cos3t$
$\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2$ = $(-3\sin t+3\sin 3t)^2+(3\cos t-3\cos 3t)^2$
$=9\sin^2t-18\sin t\sin3t+9\sin^2 3t+9\cos^2t-18\cos t\cos 3t+9\cos^2 3t$
$=9(\sin^2 t+\cos^2 t)-18(\cos t\cos 3t+\sin t\sin 3t)+9(\sin^2 3t+\cos^2 3t)$
$=9-18\cos 2t+9$
$=18-18\cos 2t$
$=18(1-\cos 2t)$
$=18(2\sin^2 t)$
$=36\sin^2 t$
$L$ = $\int_0^{\pi}{\sqrt {36\sin^2t}}dt$
$L$ = $6\int_0^{\pi}{\sin t}dt$
$L$ = $6[-\cos t]_0^{\pi}$
$L$ = $12$
