11 The curve \(C\) has polar equation \(r = a ( 1 + \sin \theta )\) for \(- \pi < \theta \leqslant \pi\), where \(a\) is a positive constant.
- Sketch \(C\).
- Find the area of the region enclosed by \(C\).
- Show that the length of the arc of \(C\) from the pole to the point furthest from the pole is given by
$$s = ( \sqrt { } 2 ) a \int _ { - \frac { 1 } { 2 } \pi } ^ { \frac { 1 } { 2 } \pi } \sqrt { } ( 1 + \sin \theta ) \mathrm { d } \theta$$
- Show that the substitution \(u = 1 + \sin \theta\) reduces this integral for \(s\) to \(( \sqrt { } 2 ) a \int _ { 0 } ^ { 2 } \frac { 1 } { \sqrt { } ( 2 - u ) } \mathrm { d } u\). Hence evaluate \(s\).