| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Multi-part: volume and area |
| Difficulty | Standard +0.3 This is a straightforward volumes of revolution question requiring standard integration techniques. Part (i) involves integrating a simple rational function, and part (ii) applies the standard formula V = π∫y² dx. Both parts are routine applications of C3 content with no conceptual challenges beyond remembering formulas and expanding (x + 2/x)². |
| Spec | 1.08e Area between curve and x-axis: using definite integrals4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Notes |
| \(= \int_1^4 \left(x + \frac{2}{x}\right)dx = \left[\frac{1}{2}x^2 + 2\ln | x | \right]_1^4\) |
| \(= (8 + 2\ln 4) - \left(\frac{1}{2} + 0\right) = 7\frac{1}{2} + 2\ln 4\) | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Notes |
| \(= \pi\int_1^4 \left(x + \frac{2}{x}\right)^2 dx = \pi\int_1^4 \left(x^2 + 4 + 4x^{-2}\right)dx\) | M1 | |
| \(= \pi\left[\frac{1}{3}x^3 + 4x - 4x^{-1}\right]_1^4\) | M1 A1 | |
| \(= \pi\left[\left(\frac{64}{3} + 16 - 1\right) - \left(\frac{1}{3} + 4 - 4\right)\right] = 36\pi\) | M1 A1 | (9) |
# Question 7:
## Part (i):
| Answer/Working | Mark | Notes |
|---|---|---|
| $= \int_1^4 \left(x + \frac{2}{x}\right)dx = \left[\frac{1}{2}x^2 + 2\ln|x|\right]_1^4$ | M1 A1 | |
| $= (8 + 2\ln 4) - \left(\frac{1}{2} + 0\right) = 7\frac{1}{2} + 2\ln 4$ | M1 A1 | |
## Part (ii):
| Answer/Working | Mark | Notes |
|---|---|---|
| $= \pi\int_1^4 \left(x + \frac{2}{x}\right)^2 dx = \pi\int_1^4 \left(x^2 + 4 + 4x^{-2}\right)dx$ | M1 | |
| $= \pi\left[\frac{1}{3}x^3 + 4x - 4x^{-1}\right]_1^4$ | M1 A1 | |
| $= \pi\left[\left(\frac{64}{3} + 16 - 1\right) - \left(\frac{1}{3} + 4 - 4\right)\right] = 36\pi$ | M1 A1 | **(9)** |
---7. The finite region $R$ is bounded by the curve with equation $y = x + \frac { 2 } { x }$, the $x$-axis and the lines $x = 1$ and $x = 4$.\\
(i) Find the exact area of $R$.
The region $R$ is rotated completely about the $x$-axis.\\
(ii) Find the volume of the solid formed, giving your answer in terms of $\pi$.\\
\hfill \mbox{\textit{OCR C3 Q7 [9]}}