[Elective 4-4] (10 points)\\
In the rectangular coordinate system $x O y$, with the origin $O$ as the pole and the positive $x$-axis as the polar axis, establish a polar coordinate system. The polar equation of curve $C _ { 1 }$ is $\rho = 2 \sin \theta \left( \frac { \pi } { 4 } \leqslant \theta \leqslant \frac { \pi } { 2 } \right)$. Curve $C _ { 2 }$ is given (see original paper for full problem statement).