2 The curve \(C\) has polar equation
$$r = a \left( 1 - \mathrm { e } ^ { - \theta } \right)$$
where \(a\) is a positive constant and \(0 \leqslant \theta < 2 \pi\).
- Draw a sketch of \(C\).
- Show that the area of the region bounded by \(C\) and the lines \(\theta = \ln 2\) and \(\theta = \ln 4\) is
$$\frac { 1 } { 2 } a ^ { 2 } \left( \ln 2 - \frac { 13 } { 32 } \right)$$