7.
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\caption{Figure 3}
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The shaded region \(R\) is bounded by part of the curve with equation \(y = \frac { 1 } { 2 } ( x - 2 ) ^ { 2 }\), the \(x\)-axis and the \(y\)-axis, as shown in Fig. 3. The unit of length on both axes is 1 cm . A uniform solid \(S\) is made by rotating \(R\) through \(360 ^ { \circ }\) about the \(x\)-axis. Using integration,
- calculate the volume of the solid \(S\), leaving your answer in terms of \(\pi\),
- show that the centre of mass of \(S\) is \(\frac { 1 } { 3 } \mathrm {~cm}\) from its plane face.
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\caption{Figure 4}
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\end{figure}
A tool is modelled as having two components, a solid uniform cylinder \(C\) and the solid \(S\). The diameter of \(C\) is 4 cm and the length of \(C\) is 8 cm . One end of \(C\) coincides with the plane face of \(S\). The components are made of different materials. The weight of \(C\) is \(10 W\) newtons and the weight of \(S\) is \(2 W\) newtons. The tool lies in equilibrium with its axis of symmetry horizontal on two smooth supports \(A\) and \(B\), which are at the ends of the cylinder, as shown in Fig. 4. - Find the magnitude of the force of the support \(A\) on the tool.