(a) The overlapping region consists of two equal segments. Find the area of one of the segments and double it. Let \({\rm{\theta }}\) be the central angle between two radii drawn to the two points where the circles intersect. The area of the sector is \(\frac{1}{2}{\rm{\theta }}\,{r^2} = \frac{1}{2}{\rm{\theta }}\). But the function, \(A\left( b \right)\), must be in terms of b; so, \({\rm{\theta }}\) must be expressed in terms of b. Consider the right triangle (shown below) with one vertex at B, the hypotenuse...