If it is so urgent, have you not tried anything yourself? Did you not see that this is part of a circle? It is the part of a circle of radius 6 beyond radius 2. What is the area of a circle of radius 6? What is the area of a circle of radius 2? "$\displaystyle 2\le r\le 6$" is the area of the "washer" with inner radius 2 and outer radius 6. And $\displaystyle \frac{5\pi}{6}- \frac{\pi}{3}= \frac{5\pi}{6}- \frac{2\pi}{6}= \frac{3\pi}{6}= \frac{\pi}{2}$, a quarter of a circle.