The cross-section of a parabolic solar concentrator is modeled by the curve with equation

$$h=\frac{2}{3}{x}^2+7,x\ge 0$$

where $h$ is the height in decimetres and $x$ is the horizontal distance from the vertical axis. The concentrator is capped by a horizontal lid at $h=31$. The finite region $S$ is bounded by the curve, the $h$-axis ($x=0$), and the lid.

Find the exact area of the region $S$.