| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2011 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Rotation about x-axis: polynomial or root function |
| Difficulty | Moderate -0.8 This is a straightforward volume of revolution question requiring a simple sketch of a parabola and application of the standard formula V = π∫y²dx. The curve is elementary, bounds are clear (x from 0 to 2), and the integration of (x-2)⁴ is routine polynomial expansion and integration—easier than average A-level questions which typically involve more steps or conceptual challenges. |
| Spec | 1.02n Sketch curves: simple equations including polynomials4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Correct shape – touching positive x-axis | B1 [1] | Ignore intersections with axes |
| (ii) \((\pi) \int (x-2)^5 dx\) | M1 | Use \((\pi) \int y^2 dx\) & attempt integrate but expansion before integr needs 5 terms |
| \((\pi) \left[\frac{(x-2)^5}{5}\right]\) | A1 | |
| \((\pi)[0 - (-32)/5)]\) \(\frac{32\pi}{5}\) or 6.4π | M1 | Use of limits 0, 2 on their \((\pi) \int y^2 dx\) |
| A1 [4] | cao Rotation about y-axis max 1/5 |
**(i)** Correct shape – touching positive x-axis | B1 [1] | Ignore intersections with axes
**(ii)** $(\pi) \int (x-2)^5 dx$ | M1 | Use $(\pi) \int y^2 dx$ & attempt integrate but expansion before integr needs 5 terms
$(\pi) \left[\frac{(x-2)^5}{5}\right]$ | A1 |
$(\pi)[0 - (-32)/5)]$ $\frac{32\pi}{5}$ or 6.4π | M1 | Use of limits 0, 2 on their $(\pi) \int y^2 dx$
| A1 [4] | cao Rotation about y-axis max 1/5
---3 (i) Sketch the curve $y = ( x - 2 ) ^ { 2 }$.\\
(ii) The region enclosed by the curve, the $x$-axis and the $y$-axis is rotated through $360 ^ { \circ }$ about the $x$-axis. Find the volume obtained, giving your answer in terms of $\pi$.
\hfill \mbox{\textit{CAIE P1 2011 Q3 [5]}}