| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Multi-part: volume and related rates |
| Difficulty | Challenging +1.2 This is a multi-step volumes of revolution problem requiring integration, algebraic manipulation, and related rates. Part (i) involves setting up the volume integral with respect to y (requiring x in terms of y from the logarithmic equation), integrating exponential functions, and algebraic simplification to reach the given form—this is moderately challenging but follows standard C3 techniques. Part (ii) is a straightforward related rates application using the chain rule. The question requires more steps and careful algebra than average, but uses only standard C3 methods without requiring novel insight. |
| Spec | 1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.08d Evaluate definite integrals: between limits4.08d Volumes of revolution: about x and y axes |
\includegraphics{figure_9}
The diagram shows the curve with equation $y = 2 \ln(x - 1)$. The point $P$ has coordinates $(0, p)$. The region $R$, shaded in the diagram, is bounded by the curve and the lines $x = 0$, $y = 0$ and $y = p$. The units on the axes are centimetres. The region $R$ is rotated completely about the $y$-axis to form a solid.
\begin{enumerate}[label=(\roman*)]
\item Show that the volume, $V \text{ cm}^3$, of the solid is given by
$$V = \pi(e^p + 4e^{\frac{p}{2}} + p - 5).$$ [8]
\item It is given that the point $P$ is moving in the positive direction along the $y$-axis at a constant rate of $0.2 \text{ cm min}^{-1}$. Find the rate at which the volume of the solid is increasing at the instant when $p = 4$, giving your answer correct to 2 significant figures. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR C3 Q9 [13]}}