| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Volume with numerical methods |
| Difficulty | Standard +0.3 This is a straightforward volumes of revolution question with standard techniques. Part (a) requires recognizing that y² = 1 + e^(2x) simplifies the integral to a routine exponential integration. Parts (b)(i) and (b)(ii) are basic numerical methods requiring only trapezium rule application and understanding of concavity—both standard C4 topics with no novel problem-solving required. |
| Spec | 1.09f 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 | Marks | Guidance |
| \(V = \int_0^2 \pi y^2 \, dx = \int_0^2 \pi(1 + e^{2x}) \, dx\) | M1 | \(\int_0^2 \pi(1+e^{2x})\,dx\); limits must appear but may be later; condone omission of \(dx\) if intention clear |
| \(= \pi\left[x + \frac{1}{2}e^{2x}\right]_0^2\) | B1 | \(\left[x + \frac{1}{2}e^{2x}\right]\) independent of \(\pi\) and limits |
| \(= \pi(2 + \frac{1}{2}e^4 - \frac{1}{2})\) | DM1 | dependent on first M1; need both limits substituted into integral of the form \(ax + be^{2x}\); condone absence of \(\pi\) for M1 |
| \(= \frac{1}{2}\pi(3 + e^4)\) | A1 [4] | cao exact only |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x=0, y=1.4142\); \(x=2, y=7.4564\) | B1 | 1.414, 7.456 or better |
| \(A = \frac{0.5}{2}\{(1.4142 + 7.4564) + 2(1.9283 + 2.8964 + 4.5919)\}\) | M1 | correct formula seen (can be implied by correct intermediate step e.g. \(27.7038.../4\)) |
| \(= 6.926\) | A1 [3] | 6.926 or 6.93 (do not allow more dp) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| 8 strips: 6.823, 16 strips: 6.797. Trapezium rule overestimates this area, but the overestimate gets less as the number of strips increases. | B1 [1] | oe |
# Question 3:
## Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $V = \int_0^2 \pi y^2 \, dx = \int_0^2 \pi(1 + e^{2x}) \, dx$ | M1 | $\int_0^2 \pi(1+e^{2x})\,dx$; limits must appear but may be later; condone omission of $dx$ if intention clear |
| $= \pi\left[x + \frac{1}{2}e^{2x}\right]_0^2$ | B1 | $\left[x + \frac{1}{2}e^{2x}\right]$ independent of $\pi$ and limits |
| $= \pi(2 + \frac{1}{2}e^4 - \frac{1}{2})$ | DM1 | dependent on first M1; need **both** limits substituted into integral of the form $ax + be^{2x}$; condone absence of $\pi$ for M1 |
| $= \frac{1}{2}\pi(3 + e^4)$ | A1 [4] | cao exact only |
## Part (b)(i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x=0, y=1.4142$; $x=2, y=7.4564$ | B1 | 1.414, 7.456 or better |
| $A = \frac{0.5}{2}\{(1.4142 + 7.4564) + 2(1.9283 + 2.8964 + 4.5919)\}$ | M1 | correct formula seen (can be implied by correct intermediate step e.g. $27.7038.../4$) |
| $= 6.926$ | A1 [3] | 6.926 or 6.93 (do not allow more dp) |
## Part (b)(ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| 8 strips: 6.823, 16 strips: 6.797. Trapezium rule overestimates this area, but the overestimate gets less as the number of strips increases. | B1 [1] | oe |
---3 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]{ce44db53-2ec8-497b-a1d5-a8adf85e3929-3_656_736_482_665}
\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 Q3 [9]}}