For $m > 0$, the region bounded by the parabola $y = x^{2}$ and the line $y = mx$ has area $\frac{4}{3}$ square units. What is $m$?

Explanation

1. 01

   $x^{2} = mx$ at $x = 0$ and $x = m$; between them the line lies above the parabola.

2. 02

   $\int_0^{m} (mx - x^{2})\,dx = \frac{m^{3}}{2} - \frac{m^{3}}{3} = \frac{m^{3}}{6}$.

3. 03

   $\frac{m^{3}}{6} = \frac{4}{3}$ gives $m^{3} = 8$, so $m = 2$.