A2 June 2019 Paper 1 Q4
4 In this question you must show detailed reasoning.
Fig. 4 shows the region bounded by the curve \(y = \sec\frac{1}{2}x\), the \(x\)-axis, the \(y\)-axis and the line \(x = \frac{1}{2}\pi\).
This region is rotated through \(2\pi\) radians about the \(x\)-axis.
Find, in exact form, the volume of the solid of revolution generated. [3]
| Scheme | Marks | AO |
|---|---|---|
| DR \(\displaystyle V = \int_0^{\frac{\pi}{2}} \pi\sec^2\frac{1}{2}x\,\mathrm{d}x\) | B1 | 1.1b |
| \(= \pi\left[2\tan\dfrac{1}{2}x\right]_0^{\frac{\pi}{2}}\) | B1 | 1.1b |
| \(= \pi(2\tan\frac{1}{4}\pi - 0) = 2\pi\) | B1cao | 1.1b |
| [3] |
Notes
B1: correct integral and limits
B1: \(2\tan\frac{1}{2}x\); condone \(\pi\) missing
B1cao: unsupported B0