The resulting integral is a mess. Are you sure you are expected to do it by hand?
$\begin{align*}
&S = 2\pi \displaystyle \int \limits_0^{\pi/2}~\sqrt{2-2\cos(t)}(t-\sin(t))~dt = \\ \\
&\left . \dfrac{4}{3} \pi \sqrt{2-2 \cos (t)} \left(\cos (t)-3 t \cot \left(\dfrac{t}{2}\right)+5\right)\right|_0^{\pi/2} = \\ \\
&\dfrac{2}{3} \sqrt{2} (10-3 \pi ) \pi
\end{align*}$