Question 4 - A-Level Maths

The region \(T\) is bounded by the \(x\)-axis, the line \(y = k x\) for \(a \leqslant x \leqslant 3 a\), the line \(x = a\) and the line \(x = 3 a\), where \(k\) and \(a\) are positive constants. A uniform frustum of a cone is formed by rotating \(T\) about the \(x\)-axis. Find the \(x\)-coordinate of the centre of mass of this frustum.
A uniform lamina occupies the region (shown in Fig. 4) bounded by the \(x\)-axis, the curve \(y = 16 \left( 1 - x ^ { - \frac { 1 } { 3 } } \right)\) for \(1 \leqslant x \leqslant 8\) and the line \(x = 8\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{86d79489-aec1-4c94-bef6-45b007f818a0-4_368_519_1439_772} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
1. Find the coordinates of the centre of mass of this lamina. A hole is made in the lamina by cutting out a circular disc of area 5 square units. This causes the centre of mass of the lamina to move to the point \(( 5,3 )\).
2. Find the coordinates of the centre of the hole.