What is the area of the region in the first quadrant bounded by the curve $y = \sqrt{x}$, the line $y = x - 2$ and the $x$-axis?

Explanation

1. 01

   The curve meets the line where $\sqrt{x} = x - 2$, at $(4, 2)$; the line meets the $x$-axis at $(2, 0)$.

2. 02

   Horizontal strips run from the curve $x = y^{2}$ to the line $x = y + 2$: $\int_0^{2} (y + 2 - y^{2})\,dy = 2 + 4 - \frac{8}{3} = \frac{10}{3}$ square units.