The diagram shows the curve \(y = x^{\frac{1}{2}} \ln x\). The shaded region between the curve, the x-axis and the line \(x = e\) is denoted by \(R\).

(i) Find the equation of the tangent to the curve at the point where \(x = 1\), giving your answer in the form \(y = mx + c\).

(ii) Find by integration the volume of the solid obtained when the region \(R\) is rotated completely about the x-axis. Give your answer in terms of \(\pi\) and \(e\).