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 \(\frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 3 } { 2 } \pi\). {www.cie.org.uk} after the live examination series.
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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 $\frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 3 } { 2 } \pi$.

{www.cie.org.uk} after the live examination series.\\

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\hfill \mbox{\textit{CAIE FP1 2015 Q11 OR}}