Using horizontal strips, which integral gives the area of the region bounded by the parabola $x^{2} = 2y$ and the line $y = 2$?

Explanation

1. 01

   $x^{2} = 2y$ gives $x = \pm\sqrt{2y}$, so a strip at height $y$ runs from $-\sqrt{2y}$ to $\sqrt{2y}$, length $2\sqrt{2y}$.

2. 02

   The strips run from the vertex, $y = 0$, to $y = 2$: $\int_0^{2} 2\sqrt{2y}\,dy = \frac{16}{3}$.