What is the area of the marked region?



The marked region is symmetric and it intersects its symmetry axis in a segment.
We consider the square that has this segment as diagonal.
In the square, the marked region is the overlap of two quarter circles with radius \(\dfrac{1}{2}\).
The quarter circles have radius \(\dfrac{1}{2}\) and hence area \(\dfrac{\pi}{16}\).
The union of the two quarter circles is the square, which has area \(\dfrac{1}{4}\).
So the marked region has area \[2\cdot\dfrac{\pi}{16} - \dfrac{1}{4} = \dfrac{\pi}{8} - \dfrac{1}{4}\,.\]