| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2013 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Volume with numerical methods |
| Difficulty | Standard +0.3 Part (a) is a standard volume of revolution requiring the formula V = π∫y² dx, which simplifies nicely since y² = 1 + e^(2x) integrates directly to x + ½e^(2x). Part (b) involves routine trapezium rule application and conceptual understanding of numerical methods convergence. This is slightly easier than average as the integration in (a) is straightforward and part (b) tests standard numerical methods without requiring novel problem-solving. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.09f Trapezium rule: numerical integration4.08d Volumes of revolution: about x and y axes |
| \(x\) | 0 | 0.5 | 1 | 1.5 | 2 |
| \(y\) | 1.9283 | 2.8964 | 4.5919 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(n(4,4) = \binom{8}{4} = 70\) | M1 | For method using combinations |
| \(= 70\) | A1 | [2] |
# Question 4:
| Answer | Mark | Guidance |
|--------|------|----------|
| $n(4,4) = \binom{8}{4} = 70$ | M1 | For method using combinations |
| $= 70$ | A1 | [2] |
---4 Fig. 4 shows the curve $y = \sqrt { 1 + \mathrm { e } ^ { 2 x } }$, and the region between the curve, the $x$-axis, the $y$-axis and the line $x = 2$.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{9bceee25-35bd-448b-a4a2-1a5667be5f11-02_650_727_1176_653}
\captionsetup{labelformat=empty}
\caption{Fig. 4}
\end{center}
\end{figure}
\begin{enumerate}[label=(\alph*)]
\item Find the exact volume of revolution when the shaded region is rotated through $360 ^ { \circ }$ about the $x$-axis.
\item \begin{enumerate}[label=(\roman*)]
\item Complete the table of values, and use the trapezium rule with 4 strips to estimate the area of the shaded region.
\begin{center}
\begin{tabular}{ | l | l | l | l | l | l | }
\hline
$x$ & 0 & 0.5 & 1 & 1.5 & 2 \\
\hline
$y$ & & 1.9283 & 2.8964 & 4.5919 & \\
\hline
\end{tabular}
\end{center}
\item The trapezium rule for $\int _ { 0 } ^ { 2 } \sqrt { 1 + \mathrm { e } ^ { 2 x } } \mathrm {~d} x$ with 8 and 16 strips gives 6.797 and 6.823, although not necessarily in that order. Without doing the calculations, say which result is which, explaining your reasoning.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR MEI C4 2013 Q4 [8]}}