6.
\begin{figure}[h]
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\caption{Figure 4}
\includegraphics[alt={},max width=\textwidth]{25b3ece7-69ed-4ec4-a6c7-4cd83ec2cc5e-09_515_1015_319_477}
\end{figure}
The shaded region \(R\) is bounded by the curve with equation \(y = \frac { 1 } { 2 x ^ { 2 } }\), the \(x\)-axis and the lines \(x = 1\) and \(x = 2\), as shown in Figure 4. The unit of length on each axis is 1 m . A uniform solid \(S\) has the shape made by rotating \(R\) through \(360 ^ { \circ }\) about the \(x\)-axis.
- Show that the centre of mass of \(S\) is \(\frac { 2 } { 7 } \mathrm {~m}\) from its larger plane face.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 5}
\includegraphics[alt={},max width=\textwidth]{25b3ece7-69ed-4ec4-a6c7-4cd83ec2cc5e-09_616_431_1420_778}
\end{figure}
A sporting trophy \(T\) is a uniform solid hemisphere \(H\) joined to the solid \(S\). The hemisphere has radius \(\frac { 1 } { 2 } \mathrm {~m}\) and its plane face coincides with the larger plane face of \(S\), as shown in Figure 5. Both \(H\) and \(S\) are made of the same material. - Find the distance of the centre of mass of \(T\) from its plane face.