Find the volume of the figure shown. The cross-section of the figure is a regular 6-sided polygon. The area of the polygon can be computed as a function of the length of a line segment from the center of the 6-sided polygon to the midpoint of one of its sides and is given by $6 x^2\tan\left(\frac{\pi}{6}\right)$ where $x$ is the length of the bisector of one of the sides (shown in black on the cross-section graph). A formula similar to the cylindrical shells formula will then provide the volume of the figure. Simply replace $\pi$ in the formula $$V=2\pi\int x f(x) dx$$ with $6 \tan\left(\frac{\pi}{6}\right)$ to find the volume of the solid shown where for this solid $$f(x)=\begin{cases}x&x\le 6\\ 12-x&6<x\le 12\end{cases}$$ for $x=0$ to $12$.