What is the area of the marked region?



The marked region is the complement in the big semicircle of the two small semicircles.
The big semicircle has diameter \(1\) and hence area \(\dfrac{\pi}{8}\).
The diameters of the small semicircles are \(d\) and \(d'\) such that \[d'=1-d\,.\]
The areas of the two small semicircles are \(\dfrac{\pi}{8}d^2\) and \(\dfrac{\pi}{8}(1-d)^2\).
So the area of the marked region is \[\dfrac{\pi}{8}-\dfrac{\pi}{8}d^2- \dfrac{\pi}{8}(1-d)^2=\dfrac{\pi}{4}d(1-d)=\dfrac{\pi}{4}dd'\,.\]
Special case: If \(d=d'\), the marked region has half the area with respect to the big semicircle.