The curve \(C\) has polar equation \(r=a(1+\sin \theta)\) for \(-\pi\lt \theta \leqslant \pi\), where \(a\) is a positive constant.
(i) Sketch \(C\).

(ii) Find the area of the region enclosed by \(C\).
(iii) Show that the length of the arc of \(C\) from the pole to the point furthest from the pole is given by
\(s=(\sqrt{ } 2) a \int_{-\frac{1}{2} \pi}^{\frac{1}{2} \pi} \sqrt{ }(1+\sin \theta) \mathrm{d} \theta\).
(iv) 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 \(s\).