What is the area of the region in the first quadrant bounded by the $x$-axis, the line $y = x$ and the circle $x^{2} + y^{2} = 8$?

Explanation

1. 01

   The line meets the circle at $(2, 2)$. Split at $x = 2$: $\int_0^{2} x\,dx + \int_2^{2\sqrt{2}} \sqrt{8 - x^{2}}\,dx$.

2. 02

   $2 + (\pi - 2) = \pi$ square units: an eighth of the circle, $\frac{8\pi}{8}$.