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