The line $y = c$ divides the region bounded by the parabola $y = x^{2}$ and the line $y = 4$ into two parts of equal area. What is $c$?

Explanation

1. 01

   The part below $y = c$ is bounded by $y = x^{2}$ and $y = c$, so its area is $\frac{4}{3}c^{3/2}$; the whole region, with $c = 4$, has $\frac{32}{3}$.

2. 02

   $\frac{4}{3}c^{3/2} = \frac{16}{3}$ gives $c^{3/2} = 4$, so $c = 4^{2/3} = 2\sqrt[3]{2}$.