Problem

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

$$y=10xe^{-\frac12x}\qquad 0\le x\le 10$$

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

$$V=k\int_0^{10}x^2e^{-x}\,dx$$

where $k$ is a constant to be found.

(b) Find $\int x^2e^{-x}\,dx$

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),

(c) find the density of this exercise weight. Give your answer in grams per cm$^3$ to 3 significant figures.