6 The equation of a curve in polar coordinates is \(r = \ln ( 1 + \sin \theta )\) for \(\alpha \leqslant \theta \leqslant \beta\) where \(\alpha\) and \(\beta\) are non-negative angles. The curve consists of a single closed loop through the pole.
- By solving the equation \(r = 0\), determine the smallest possible values of \(\alpha\) and \(\beta\).
- Find the area enclosed by the curve, giving your answer to 4 significant figures.
- Hence, by considering the value of \(r\) at \(\theta = \frac { \alpha + \beta } { 2 }\), show that the loop is not circular.