What is the area of the part of the circle $x^{2} + y^{2} = 4$ to the right of the line $x = 1$?

Explanation

1. 01

   $2\int_1^{2} \sqrt{4 - x^{2}}\,dx$, using $\int \sqrt{4 - x^{2}}\,dx = \frac{x}{2}\sqrt{4 - x^{2}} + 2\sin^{-1}\frac{x}{2}$.

2. 02

   Between the limits: $\pi - \left(\frac{\sqrt{3}}{2} + \frac{\pi}{3}\right) = \frac{2\pi}{3} - \frac{\sqrt{3}}{2}$; doubled, $\frac{4\pi}{3} - \sqrt{3}$.