| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2015 |
| Session | June |
| Marks | 4 |
| Topic | Volumes of Revolution |
| Type | Rotation about x-axis: polynomial or root function |
| Difficulty | Moderate -0.8 This is a straightforward application of the volume of revolution formula V = π∫y² dx with a simple polynomial function. The integration of x^6 between clear limits requires only basic technique with no conceptual challenges or multi-step reasoning, making it easier than average. |
| Spec | 4.08d Volumes of revolution: about x and y axes |
**Question 4:**
$$\pi\int_1^2 x^6\,dx = \pi\left[\frac{x^7}{7}\right]_1^2 = \pi\left(\frac{128}{7}-\frac{1}{7}\right) = \frac{127}{7}\pi \quad (= 57.0 \text{ to 3 sf})$$
- **B1**: State or imply correct formula for volume of revolution
- **M1**: Attempt integration to obtain $kx^7$
- **M1**: Attempt use of limits in any integration attempt (i.e. increase in power by 1); must be correct order and subtraction (M0M1 is possible)
- **A1**: Obtain $\frac{127\pi}{7}$, or 57.0 or better (allow $\pi\frac{127}{7}$)
**Total: [4]**4 Find the volume of the solid generated when the region bounded by the $x$-axis, $x = 1 , x = 2$ and the curve given by $y = x ^ { 3 }$ is rotated through $360 ^ { \circ }$ about the $x$-axis.
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2015 Q4 [4]}}