In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figure 2 shows the curve with equation $$\[ y=10 x \mathrm{e}^{-\frac{1}{2} x} \quad 0 \leqslant x \leqslant 10 \]$$ The finite region $$\(R\)$$, shown shaded in Figure 2, is bounded by the curve, the $$\(x\)$$-axis and the line with equation $$\(x=10\)$$ The region $$\(R\)$$ is rotated through $$\(2 \pi\)$$ radians about the $$\(x\)$$-axis to form a solid of revolution. (a) Show that the volume, $$\(V\)$$, of this solid is given by $$\[ V=k \int_{0}^{10} x^{2} \mathrm{e}^{-x} \mathrm{~d} x \]$$ where $$\(k\)$$ is a constant to be found. (2) (b) Find $$\(\int x^{2} e^{-x} d x\)$$ (3) Figure 3 represents an exercise weight formed by joining two of these solids together. The exercise weight has mass $$\(5 \mathrm{~kg}\)$$ and is $$\(20 \mathrm{~cm}\)$$ long. Given that $$\[ \text { density }=\frac{\text { mass }}{\text { volume }} \]$$ and using your answers to part (a) and part (b), (c) find the density of this exercise weight. Give your answer in grams per $$\(\mathrm{cm}^{3}\)$$ to 3 significant figures. (5)

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