What is the area of the marked region?



The marked region is the complement of a circle with area \(\dfrac{\pi}{4}\) of two regions with area \(\dfrac{\pi}{2} - \dfrac{1}{4}\), see .
Hence its area is \[\dfrac{\pi}{4} - 2\cdot \left(\dfrac{\pi}{8} - \dfrac{1}{4}\right) = \dfrac{1}{2} \,.\]