The diagram shows the curve \(y = e^{-\frac{1}{2}x} \sqrt{1 + 2x}\) and its maximum point \(M\). The shaded region between the curve and the axes is denoted by \(R\).

(i) Find the \(x\)-coordinate of \(M\).

(ii) 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 \(e\).