3 In this question, give your answers in an exact form.
The region \(R _ { 1 }\) (shown in Fig. 3) is bounded by the \(x\)-axis, the lines \(x = 1\) and \(x = 5\), and the curve \(y = \frac { 1 } { x }\) for \(1 \leqslant x \leqslant 5\).
- A uniform solid of revolution is formed by rotating the region \(R _ { 1 }\) through \(2 \pi\) radians about the \(x\)-axis. Find the \(x\)-coordinate of the centre of mass of this solid.
- Find the coordinates of the centre of mass of a uniform lamina occupying the region \(R _ { 1 }\).
\begin{figure}[h]
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\caption{Fig. 3}
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The region \(R _ { 2 }\) is bounded by the \(y\)-axis, the lines \(y = 1\) and \(y = 5\), and the curve \(y = \frac { 1 } { x }\) for \(\frac { 1 } { 5 } \leqslant x \leqslant 1\). The region \(R _ { 3 }\) is the square with vertices \(( 0,0 ) , ( 1,0 ) , ( 1,1 )\) and \(( 0,1 )\). - Write down the coordinates of the centre of mass of a uniform lamina occupying the region \(R _ { 2 }\).
- Find the coordinates of the centre of mass of a uniform lamina occupying the region consisting of \(R _ { 1 } , R _ { 2 }\) and \(R _ { 3 }\) (shown shaded in Fig. 3).