A solid has base region S in the first quadrant bounded above by the piecewise curve
$y = \begin{cases}
x, & 0\le x\le 1,
\sqrt{x}, & 1< x\le 4
\end{cases}$
and below by the x-axis. Cross sections taken perpendicular to the x-axis are rectangles whose height is three times their base width. Find the exact volume of the solid.