For $c > 0$, what is the area of the region bounded by the parabola $y = x^{2}$ and the line $y = c$?

Explanation

1. 01

   $y = c$ meets $y = x^{2}$ at $x = \pm\sqrt{c}$ and lies above it between them.

2. 02

   $\int_{-\sqrt{c}}^{\sqrt{c}} (c - x^{2})\,dx = 2c^{3/2} - \frac{2}{3}c^{3/2} = \frac{4}{3}c^{3/2}$ square units.