The curve \(C\) has polar equation

\(r=a(1-\cos\theta),\qquad 0\leq \theta\lt 2\pi.\)

Sketch \(C\).

Find the area of the region enclosed by the arc of \(C\) for which \(\frac{1}{2}\pi\leq\theta\leq\frac{3}{2}\pi\), the half-line \(\theta=\frac{1}{2}\pi\), and the half-line \(\theta=\frac{3}{2}\pi\).

Show that

\(\left(\frac{ds}{d\theta}\right)^2=4a^2\sin^2\left(\frac{\theta}{2}\right),\)

where \(s\) denotes arc length, and find the length of the arc of \(C\) for which \(\frac{1}{2}\pi\leq\theta\leq\frac{3}{2}\pi\).