Answer
$S\approx 4.7394$
Work Step by Step
$x$ = $t\sin t$
$y$ = $t\cos t$
$\frac{dx}{dt}$ = $t\cos t+\sin t$
$\frac{dy}{dt}$ = $-t\sin t+\cos t$
$\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2$ = $(t\cos t+\sin t)^2+(-t\sin t+\cos t)^2$
$=t^2\cos^2t+2t\sin t\cos t+\sin^2t+t^2\sin^2t-2t\sin t\cos t+\cos^2t$ $=t^2+1$
$S$ = $\int2{\pi}yds$
$S$ = $\int_0^{\frac{\pi}{2}}2{\pi}t\cos t\sqrt {t^2+1}dt$ $\approx$ $4.7394$
