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The diagram shows the curve \(y = \mathrm { e } ^ { - \frac { 1 } { 2 } x } \sqrt { } ( 1 + 2 x )\) and its maximum point \(M\). The shaded region between the curve and the axes is denoted by \(R\).
- Find the \(x\)-coordinate of \(M\).
- Find by integration the volume of the solid obtained when \(R\) is rotated completely about the \(x\)-axis. Give your answer in terms of \(\pi\) and e.