The region $$\(R\)$$, shown shaded in Figure 1, is bounded by the curve with equation $$\(y=\frac{1}{x}\)$$, the line with equation $$\(x=1\)$$, the positive $$\(x\)$$-axis and the line with equation $$\(x=a\)$$ where $$\(a> 1\)$$ A uniform solid $$\(S\)$$ is formed by rotating $$\(R\)$$ through $$\(2 \pi\)$$ radians about the $$\(x\)$$-axis. (a) Show that the volume of $$\(S\)$$ is $$\[ \pi\left(1-\frac{1}{a}\right) \]$$ (3) (b) Find the $$\(x\)$$ coordinate of the centre of mass of $$\(S\)$$. (5)

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