C4 January 2006 Q4
4.
Figure 1 shows the finite shaded region, \(R\), which is bounded by the curve \(y = x\mathrm{e}^x\), the line \(x = 1\), the line \(x = 3\) and the \(x\)-axis.
The region \(R\) is rotated through 360 degrees about the \(x\)-axis.
Use integration by parts to find an exact value for the volume of the solid generated. (8)
| Scheme | Marks |
|---|---|
| Attempts \(V = \pi\displaystyle\int x^2\mathrm{e}^{2x}\,\mathrm{d}x\) | M1 |
| \(= \pi\left[\dfrac{x^2\mathrm{e}^{2x}}{2} - \displaystyle\int x\mathrm{e}^{2x}\,\mathrm{d}x\right]\) (M1 needs parts in the correct direction) | M1 A1 |
| \(= \pi\left[\dfrac{x^2\mathrm{e}^{2x}}{2} - \left(\dfrac{x\mathrm{e}^{2x}}{2} - \displaystyle\int \frac{\mathrm{e}^{2x}}{2}\,\mathrm{d}x\right)\right]\) (M1 needs second application of parts) | M1 A1ft |
| \(= \pi\left[\dfrac{x^2\mathrm{e}^{2x}}{2} - \left(\dfrac{x\mathrm{e}^{2x}}{2} - \dfrac{\mathrm{e}^{2x}}{4}\right)\right]\) | A1 cao |
| Substitutes limits 3 and 1 and subtracts to give… [dep. on second and third Ms] | dM1 |
| \(= \pi\left[\frac{13}{4}\mathrm{e}^6 - \frac{1}{4}\mathrm{e}^2\right]\) or any correct exact equivalent. | A1 |
| (8) | |
| (8 marks) |
Notes
M1A1ft refers to candidates \(\displaystyle\int x\mathrm{e}^{2x}\,\mathrm{d}x\), but dependent on prev. M1
[Omission of \(\pi\) loses first and last marks only]