What is the area of the marked region?



The marked region is the complement of the circle with area \(2\pi\) of an region with area \(\dfrac{4\pi}{3} - \dfrac{\sqrt{3}}{2}\),see
Hence its area is \[2\pi - \dfrac{4\pi}{3} + \dfrac{\sqrt{3}}{2} = \dfrac{2\pi}{3} + \dfrac{\sqrt{3}}{2}\,.\]