The curve \(C\) has polar equation \(r = a ( 1 - \cos \theta )\) for \(0 \leqslant \theta < 2 \pi\). Sketch \(C\). Find the area of the region enclosed by the arc of \(C\) for which \(\frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 3 } { 2 } \pi\), the half-line \(\theta = \frac { 1 } { 2 } \pi\) and the half-line \(\theta = \frac { 3 } { 2 } \pi\). Show that $$\left( \frac { \mathrm { d } s } { \mathrm {~d} \theta } \right) ^ { 2 } = 4 a ^ { 2 } \sin ^ { 2 } \left( \frac { 1 } { 2 } \theta \right) ,$$ where \(s\) denotes arc length, and find the length of the arc of \(C\) for which \(\frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 3 } { 2 } \pi\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.
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