M3 January 2011 Q3
3.
The region \(R\) is bounded by the curve with equation \(y = \mathrm{e}^x\), the line \(x = 1\), the line \(x = 2\) and the \(x\)-axis as shown in Figure 2. A uniform solid \(S\) is formed by rotating \(R\) through \(2\pi\) about the \(x\)-axis.
(a) Show that the volume of \(S\) is \(\tfrac{1}{2}\pi\left(\mathrm{e}^4 - \mathrm{e}^2\right)\). (4)
(b) Find, to 3 significant figures, the \(x\)-coordinate of the centre of mass of \(S\). (6)
| Scheme | Marks |
|---|---|
| \(\text{Vol } = \pi\displaystyle\int_1^2 y^2\,\mathrm{d}x = \pi\int_1^2 \mathrm{e}^{2x}\,\mathrm{d}x\) | M1 |
| \(= \dfrac{1}{2}\pi\left[\mathrm{e}^{2x}\right]_1^2\) | M1 A1 |
| \(= \dfrac{1}{2}\pi\left[\mathrm{e}^4 - \mathrm{e}^2\right]\) | A1 |
| (4) |
| Scheme | Marks |
|---|---|
| C of M \(= \dfrac{\displaystyle\int_1^2 \pi y^2 x\,\mathrm{d}x}{\text{vol}}\) | |
| \(\displaystyle\int_1^2 \mathrm{e}^{2x}x\,\mathrm{d}x = \left[\dfrac{1}{2}x\mathrm{e}^{2x}\right]_1^2 - \int_1^2 \dfrac{1}{2}\mathrm{e}^{2x}\,\mathrm{d}x\) | M1 A1 |
| \(= \left[\dfrac{1}{2}x\mathrm{e}^{2x}\right]_1^2 - \left[\dfrac{1}{4}\mathrm{e}^{2x}\right]_1^2\) | M1 |
| \(= \dfrac{1}{2} \times 2\mathrm{e}^4 - \dfrac{1}{2} \times 1\mathrm{e}^2 - \left(\dfrac{1}{4}\mathrm{e}^4 - \dfrac{1}{4}\mathrm{e}^2\right)\) | |
| \(= \left(\dfrac{3}{4}\mathrm{e}^4 - \dfrac{1}{4}\mathrm{e}^2\right)\) | A1 |
| C of M \(= \dfrac{\pi\left(\dfrac{3}{4}\mathrm{e}^4 - \dfrac{1}{4}\mathrm{e}^2\right)}{\dfrac{1}{2}\pi\left(\mathrm{e}^4 - \mathrm{e}^2\right)} = 1.656\ldots\) \(= 1.66\) (3 sf) | M1 A1 |
| (6) | |
| (10 marks) |