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\includegraphics[max width=\textwidth, alt={}, center]{653d57aa-7775-4063-a8c9-11c8bc964fae-2_566_606_264_772} The curve \(C\) has polar equation $$r = \theta ^ { \frac { 1 } { 2 } } \mathrm { e } ^ { \theta ^ { 2 } / \pi }$$ where \(0 \leqslant \theta \leqslant \pi\). The area of the finite region bounded by \(C\) and the line \(\theta = \beta\) is \(\pi\) (see diagram). Show that $$\beta = ( \pi \ln 3 ) ^ { \frac { 1 } { 2 } }$$