Which expression gives the area of the region in the first quadrant bounded by the curve $y = x^{2}$, the line $x + y = 2$ and the $x$-axis?

Explanation

1. 01

   $y = x^{2}$ meets $x + y = 2$ where $x^{2} + x - 2 = 0$, at $x = 1$ in the first quadrant; the line meets the $x$-axis at $x = 2$.

2. 02

   The top edge is the parabola for $0 \le x \le 1$ and the line for $1 \le x \le 2$: $\int_0^{1} x^{2}\,dx + \int_1^{2} (2 - x)\,dx = \frac{1}{3} + \frac{1}{2} = \frac{5}{6}$.