Minimizing Area

The minimum occurs when the base of the circular segment is equal to the horizontal lengths of the other two shaded areas, since any deviation from it will increase total area. Hence, $y = \sqrt{3-({\frac{\sqrt{3}}{2}})^2} = \frac{3}{2}$

To see how any deviation will increase the total area, first draw the tangent at the intersection where $x=\frac{\sqrt{3}}{2}$ .

Then examine closely what happens if the line is moved. To a first order approximation (along the tangent line), there is no net change in area, but in the 2nd order approximation (along the circular arc), there is a net increase in area.