4 A curve is defined parametrically by

$$x = t - \frac { 1 } { 2 } \sin 2 t \quad \text { and } \quad y = \sin ^ { 2 } t$$

The arc of the curve joining the point where $t = 0$ to the point where $t = \pi$ is rotated through one complete revolution about the $x$-axis. The area of the surface generated is denoted by $S$.\\
(i) Show that

$$S = a \pi \int _ { 0 } ^ { \pi } \sin ^ { 3 } t \mathrm {~d} t$$

where the constant $a$ is to be found.\\

(ii) Using the result $\sin 3 t = 3 \sin t - 4 \sin ^ { 3 } t$, find the exact value of $S$.\\

\hfill \mbox{\textit{CAIE FP1 2018 Q4}}