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\includegraphics[max width=\textwidth, alt={}, center]{b8ff33d4-dfe5-4067-855e-86d5765cc249-2_454_770_1628_635} The diagram shows the curve \(y = \frac { 6 } { \sqrt { 3 x + 1 } }\). The shaded region is bounded by the curve and the lines \(x = 2 , x = 9\) and \(y = 0\).

1. Show that the area of the shaded region is \(4 \sqrt { 7 }\) square units.
2. The shaded region is rotated completely about the \(x\)-axis. Show that the volume of the solid produced can be written in the form \(k \ln 2\), where the exact value of the constant \(k\) is to be determined.