What is the area of the marked region?



The region is the overlap of two quarters of circle.
Each quarter of circle has area \(\dfrac{\pi}{4}\).
The union of the two quarter of circles is the full square, which has area \(1\).
Calling \(A\) the area of the overlap, we have \[\dfrac{\pi}{4}+\dfrac{\pi}{4}-A=1\,.\] We deduce \[A= \dfrac{\pi}{2} - 1\,.\]