What is the area of the region bounded by the parabola $y = x^{2}$ and the line $y = x + 2$?

Explanation

1. 01

   $x^{2} = x + 2$ gives $x = -1$ and $x = 2$; between them the line lies above the parabola.

2. 02

   $\int_{-1}^{2} (x + 2 - x^{2})\,dx = \left[\frac{x^{2}}{2} + 2x - \frac{x^{3}}{3}\right]_{-1}^{2} = \frac{10}{3} + \frac{7}{6} = \frac{9}{2}$ square units.