| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2008 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Rotation about x-axis: rational or reciprocal function |
| Difficulty | Moderate -0.3 Part (a) is a straightforward integration by inspection/substitution of a linear function raised to a power. Part (b) is a standard volume of revolution question requiring the formula V = π∫y²dx with a simple function (x^(-1/2))² = x^(-1), leading to a routine logarithm answer. Both parts are textbook exercises with no problem-solving required, making this slightly easier than average for C3. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Obtain integral of form \(k(3x + 7)^{10}\) | M1 | any constant \(k\) |
| Obtain (unsimplified) \(\frac{1}{10}x(3x + 7)^{10}\) | A1 | or equiv |
| Obtain (simplified) \(\frac{1}{30}(3x + 7)^{10} + c\) | A1 3 | |
| (b) State \(\pi(\frac{-\ln x}{x})^2 dx\) | B1 | or equiv involving \(x\); condone no \(dx\) |
| Integrate to obtain \(k \ln x\) | M1 | any constant \(k\) involving \(\pi\) or not; or equiv such as \(k \ln 4 x\) or \(k \ln 2x\) |
| Obtain \(\frac{1}{4}\pi \ln x\) or \(\frac{1}{4}\ln x\) or \(\frac{1}{4}\pi \ln 4x\) or \(\frac{1}{4}\ln 4x\) | A1 | |
| Show use of the \(\log a - \log b\) property | M1 | not dependent on earlier marks |
| Obtain \(\frac{1}{4}\pi \ln 2\) | A1 5 | or similarly simplified equiv |
**(a)** Obtain integral of form $k(3x + 7)^{10}$ | M1 | any constant $k$
Obtain (unsimplified) $\frac{1}{10}x(3x + 7)^{10}$ | A1 | or equiv
Obtain (simplified) $\frac{1}{30}(3x + 7)^{10} + c$ | A1 3 |
**(b)** State $\pi(\frac{-\ln x}{x})^2 dx$ | B1 | or equiv involving $x$; condone no $dx$
Integrate to obtain $k \ln x$ | M1 | any constant $k$ involving $\pi$ or not; or equiv such as $k \ln 4 x$ or $k \ln 2x$
Obtain $\frac{1}{4}\pi \ln x$ or $\frac{1}{4}\ln x$ or $\frac{1}{4}\pi \ln 4x$ or $\frac{1}{4}\ln 4x$ | A1 |
Show use of the $\log a - \log b$ property | M1 | not dependent on earlier marks
Obtain $\frac{1}{4}\pi \ln 2$ | A1 5 | or similarly simplified equiv
---5
\begin{enumerate}[label=(\alph*)]
\item Find $\int ( 3 x + 7 ) ^ { 9 } \mathrm {~d} x$.
\item \\
\includegraphics[max width=\textwidth, alt={}, center]{32f90420-e1eb-47ab-b588-e3806b64813f-3_537_881_402_671}
The diagram shows the curve $y = \frac { 1 } { 2 \sqrt { x } }$. The shaded region is bounded by the curve and the lines $x = 3 , x = 6$ and $y = 0$. The shaded region is rotated completely about the $x$-axis. Find the exact volume of the solid produced, simplifying your answer.
\end{enumerate}
\hfill \mbox{\textit{OCR C3 2008 Q5 [8]}}