Points A, B, and C lie on a line, in that order, with A B=8 and B C=2 . B is rotated 20^{\circ} counterclockwise about A to a point B^{\prime}, tracing out an arc R_{1}. C is then rotated 20^{\circ} clockwise about A to a point C^{\prime}, tracing out an arc R_{2}. What is the area of the region bounded by arc R_{1}, segment B^{\prime} C, arc R_{2}, and segment C^{\prime} B?