4 The region \(R\) is bounded by the \(x\)-axis, the \(y\)-axis, the curve \(y = \mathrm { e } ^ { - x }\) and the line \(x = k\), where \(k\) is a positive constant.

1. The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution. Find the \(x\)-coordinate of the centre of mass of this solid, and show that it can be written in the form $$\frac { 1 } { 2 } - \frac { k } { \mathrm { e } ^ { 2 k } - 1 } .$$
2. The solid in part (i) is placed with its larger circular face in contact with a rough plane inclined at \(60 ^ { \circ }\) to the horizontal, as shown in Fig. 4, and you are given that no slipping occurs. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5a0df44f-f8f0-44d4-b2f6-70a5314706f9-5_508_483_712_790} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure} Show that, whatever the value of \(k\), the solid will not topple.
3. A uniform lamina occupies the region \(R\). Find, in terms of \(k\), the coordinates of the centre of mass of this lamina. \section*{END OF QUESTION PAPER}