- In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
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\caption{Figure 2}
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\caption{Figure 3}
\end{figure}
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.
- 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.
- Find \(\int x ^ { 2 } e ^ { - x } d x\)
Figure 3 represents an exercise weight formed by joining two of these solids together.
The exercise weight has mass 5 kg and is 20 cm long.
Given that
$$\text { density } = \frac { \text { mass } } { \text { volume } }$$
and using your answers to part (a) and part (b), - find the density of this exercise weight. Give your answer in grams per \(\mathrm { cm } ^ { 3 }\) to 3 significant figures.