| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2005 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Volume with numerical methods |
| Difficulty | Moderate -0.3 Part (a) is a straightforward volume of revolution requiring the standard formula V = π∫y²dx with a simple function that integrates easily to give 8π. Part (b) is routine application of Simpson's rule with clear strip width. Both parts are standard textbook exercises requiring only direct application of learned techniques, making this slightly easier than average but not trivial due to the two-part structure and need for careful arithmetic. |
| Spec | 1.09f Trapezium rule: numerical integration4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| (a) State or imply \(\int \pi y^2 \, dx\) | B1 | |
| Integrate to obtain \(k \ln x\) | M1 | [any constant \(k\), involving \(\pi\) or not; or equiv such as \(k \ln 4x\)] |
| Obtain \(4\pi \ln x\) or \(4 \ln x\) | A1 | [or equiv] |
| Obtain \(4\pi \ln 5\) | A1 | Total: 4 marks [or similarly simplified equiv] |
| (b) Attempt calculation involving attempts at \(y\) values | M1 | [with each of 1, 4, 2 present at least once as coefficients] |
| Attempt \(\frac{1}{2} \times (y_0 + 4y_1 + 2y_2 + 4y_3 + y_4)\) | M1 | [with attempts at five \(y\) values] |
| Obtain \(\frac{1}{3}(\sqrt{2} + 4\sqrt{5} + 2\sqrt{10} + 4\sqrt{17} + \sqrt{26})\) | A1 | [or exact equiv or decimal equivs] |
| Obtain 12.758 | A1 | Total: 4 marks [or greater accuracy] |
**(a)** State or imply $\int \pi y^2 \, dx$ | B1 |
Integrate to obtain $k \ln x$ | M1 | [any constant $k$, involving $\pi$ or not; or equiv such as $k \ln 4x$]
Obtain $4\pi \ln x$ or $4 \ln x$ | A1 | [or equiv]
Obtain $4\pi \ln 5$ | A1 | Total: 4 marks [or similarly simplified equiv]
**(b)** Attempt calculation involving attempts at $y$ values | M1 | [with each of 1, 4, 2 present at least once as coefficients]
Attempt $\frac{1}{2} \times (y_0 + 4y_1 + 2y_2 + 4y_3 + y_4)$ | M1 | [with attempts at five $y$ values]
Obtain $\frac{1}{3}(\sqrt{2} + 4\sqrt{5} + 2\sqrt{10} + 4\sqrt{17} + \sqrt{26})$ | A1 | [or exact equiv or decimal equivs]
Obtain 12.758 | A1 | Total: 4 marks [or greater accuracy]4
\begin{enumerate}[label=(\alph*)]
\item \\
\includegraphics[max width=\textwidth, alt={}, center]{e0e2a26b-d4d6-46ea-ac12-a882f3465e5e-2_579_785_1279_721}
The diagram shows the curve $y = \frac { 2 } { \sqrt { } x }$. The region $R$, shaded in the diagram, is bounded by the curve and by the lines $x = 1 , x = 5$ and $y = 0$. The region $R$ is rotated completely about the $x$-axis. Find the exact volume of the solid formed.
\item Use Simpson's rule, with 4 strips, to find an approximate value for
$$\int _ { 1 } ^ { 5 } \sqrt { } \left( x ^ { 2 } + 1 \right) \mathrm { d } x ,$$
giving your answer correct to 3 decimal places.
\end{enumerate}
\hfill \mbox{\textit{OCR C3 2005 Q4 [8]}}