The curve \(C\) has polar equation \(r=2 \mathrm{e}^{\theta}\), for \(\frac{1}{6} \pi \leqslant \theta \leqslant \frac{1}{2} \pi\). Find
(i) the area of the region bounded by the half-lines \(\theta=\frac{1}{6} \pi, \theta=\frac{1}{2} \pi\) and \(C\),

(ii) the length of \(C\).