| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2007 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Multi-part: volume and area |
| Difficulty | Moderate -0.3 This is a straightforward volumes of revolution question requiring standard integration techniques. Part (i) involves integrating (3x+2)^{-1/2} using the reverse chain rule, and part (ii) uses the standard formula V = π∫y² dx which simplifies to a linear integration. Both parts are routine applications of C3 techniques with no problem-solving insight required, making it slightly easier than average. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Obtain integral of form \(k(3x+2)^{\frac{1}{2}}\) | M1 | any constant \(k\) |
| Obtain correct \(\frac{2}{3}(3x+2)^{\frac{3}{2}}\) | A1 | or equiv |
| Substitute limits 0 and 2 and attempt evaluation | M1 | for integral of form \(k(3x+2)^n\) |
| Obtain \(\frac{2}{3}(8^{\frac{3}{2}} - 2^{\frac{3}{2}})\) | A1 | 4 or exact equiv suitably simplified |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State or imply \(\pi\int\frac{1}{3x+2}\,dx\) or unsimplified version | B1 | allow if \(dx\) absent or wrong |
| Obtain integral of form \(k\ln(3x+2)\) | M1 | any constant \(k\) involving \(\pi\) or not |
| Obtain \(\frac{1}{3}\pi\ln(3x+2)\) or \(\frac{1}{3}\ln(3x+2)\) | A1 | |
| Show correct use of \(\ln a - \ln b\) property | M1 | |
| Obtain \(\frac{1}{3}\pi\ln 4\) | A1 | 5 or (similarly simplified) equiv |
# Question 6:
## Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Obtain integral of form $k(3x+2)^{\frac{1}{2}}$ | M1 | any constant $k$ |
| Obtain correct $\frac{2}{3}(3x+2)^{\frac{3}{2}}$ | A1 | or equiv |
| Substitute limits 0 and 2 and attempt evaluation | M1 | for integral of form $k(3x+2)^n$ |
| Obtain $\frac{2}{3}(8^{\frac{3}{2}} - 2^{\frac{3}{2}})$ | A1 | **4** or exact equiv suitably simplified |
## Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State or imply $\pi\int\frac{1}{3x+2}\,dx$ or unsimplified version | B1 | allow if $dx$ absent or wrong |
| Obtain integral of form $k\ln(3x+2)$ | M1 | any constant $k$ involving $\pi$ or not |
| Obtain $\frac{1}{3}\pi\ln(3x+2)$ or $\frac{1}{3}\ln(3x+2)$ | A1 | |
| Show correct use of $\ln a - \ln b$ property | M1 | |
| Obtain $\frac{1}{3}\pi\ln 4$ | A1 | **5** or (similarly simplified) equiv |
---6\\
\includegraphics[max width=\textwidth, alt={}, center]{1216a06e-7e14-48d7-a7ca-7acd8d71af5f-3_483_956_264_593}
The diagram shows the curve with equation $y = \frac { 1 } { \sqrt { 3 x + 2 } }$. The shaded region is bounded by the curve and the lines $x = 0 , x = 2$ and $y = 0$.\\
(i) Find the exact area of the shaded region.\\
(ii) The shaded region is rotated completely about the $x$-axis. Find the exact volume of the solid formed, simplifying your answer.
\hfill \mbox{\textit{OCR C3 2007 Q6 [9]}}