The curve \(C\) has polar equation
\(r=a\left(1-\mathrm{e}^{-\theta}\right),\)
where \(a\) is a positive constant and \(0 \leqslant \theta\lt 2 \pi\).
(i) Draw a sketch of \(C\).

(ii) Show that the area of the region bounded by \(C\) and the lines \(\theta=\ln 2\) and \(\theta=\ln 4\) is
\(\frac{1}{2} a^{2}\left(\ln 2-\frac{13}{32}\right) .\)