- A curve has parametric equations
where \(a\) is a positive constant.
$$\begin{aligned}
& x = a ( \theta - \sin \theta )
& y = a ( 1 - \cos \theta )
\end{aligned}$$
- Show that
$$\left( \frac { \mathrm { d } x } { \mathrm {~d} \theta } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} \theta } \right) ^ { 2 } = k a ^ { 2 } \sin ^ { 2 } \frac { \theta } { 2 }$$
where \(k\) is a constant to be determined.
The part of the curve from \(\theta = 0\) to \(\theta = 2 \pi\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
- Determine the area of the surface generated, giving your answer in terms of \(\pi\) and \(a\).
[0pt]
[Solutions relying on calculator technology are not acceptable.]