Figure 3 shows part of the curve $$\(C\)$$ with equation $$\(y=x^{2}+4\)$$. The shaded region $$\(R\)$$ is bounded by $$\(C\)$$, the line with equation $$\(x=1\)$$, the $$\(x\)$$-axis and the line with equation $$\(x=2\)$$ The unit of length on each axis is one centimetre. A uniform wooden solid, $$\(S\)$$, is made in the shape formed by rotating the region $$\(R\)$$ through $$\(360^{\circ}\)$$ about the $$\(x\)$$-axis. (a) Using algebraic integration, (i) show that the volume of $$\(S\)$$ is $$\(\frac{613 \pi}{15} \mathrm{~cm}^{3}\)$$ (ii) find, to 3 significant figures, the distance of the centre of mass of $$\(S\)$$ from $$\(O\)$$. (8) A solid, $$\(S_{1}\)$$, is formed by removing a solid cylinder of radius $$\(3 \mathrm{~cm}\)$$ and length $$\(1 \mathrm{~cm}\)$$ from $$\(S\)$$. A metal cylinder, of radius $$\(3 \mathrm{~cm}\)$$ and length $$\(1 \mathrm{~cm}\)$$ is placed in the resulting hole to form a new solid $$\(T\)$$, as shown in Figure 4. The axis of the metal cylinder coincides with the axis of symmetry of $$\(S_{1}\)$$. The point $$\(B\)$$ is the centre of the smaller plane face of $$\(T\)$$. The mass per unit volume of $$\(S_{1}\)$$ is $$\(M\)$$ and the mass per unit volume of the metal cylinder is $$\(5 M\)$$. (b) Find the distance of the centre of mass of $$\(T\)$$ from $$\(B\)$$. (5)

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