| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2006 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Volume requiring substitution or integration by parts |
| Difficulty | Standard +0.8 This is a standard volumes of revolution question requiring integration by parts, but the function x²e^(2x) resulting from (xe^x)² requires two applications of integration by parts to solve completely. The technique is prescribed and the setup is routine, but the algebraic manipulation across multiple integration by parts steps elevates it above average difficulty. |
| Spec | 1.08i Integration by parts4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| Attempts \(V = \pi\int x^2 e^{2x}\,dx\) | M1 | |
| \(= \pi\left[\frac{x^2 e^{2x}}{2} - \int xe^{2x}\,dx\right]\) | M1, A1 | M1 needs parts in the correct direction |
| \(= \pi\left[\frac{x^2 e^{2x}}{2} - \left(\frac{xe^{2x}}{2} - \int\frac{e^{2x}}{2}\,dx\right)\right]\) | M1, A1\(\sqrt{}\) | M1 needs second application of parts; A1\(\sqrt{}\) refers to \(\int xe^{2x}\,dx\), dependent on prev. M1 |
| \(= \pi\left[\frac{x^2 e^{2x}}{2} - \left(\frac{xe^{2x}}{2} - \frac{e^{2x}}{4}\right)\right]\) | A1 cao | |
| Substitutes limits \(3\) and \(1\) and subtracts | dM1 | Dependent on second and third Ms |
| \(= \pi\left[\frac{13}{4}e^6 - \frac{1}{4}e^2\right]\) or any correct exact equivalent | A1 | [8] |
| Omission of \(\pi\) loses first and last marks only |
## Question 4:
| Working/Answer | Marks | Guidance |
|---|---|---|
| Attempts $V = \pi\int x^2 e^{2x}\,dx$ | M1 | |
| $= \pi\left[\frac{x^2 e^{2x}}{2} - \int xe^{2x}\,dx\right]$ | M1, A1 | M1 needs parts in the correct direction |
| $= \pi\left[\frac{x^2 e^{2x}}{2} - \left(\frac{xe^{2x}}{2} - \int\frac{e^{2x}}{2}\,dx\right)\right]$ | M1, A1$\sqrt{}$ | M1 needs second application of parts; A1$\sqrt{}$ refers to $\int xe^{2x}\,dx$, dependent on prev. M1 |
| $= \pi\left[\frac{x^2 e^{2x}}{2} - \left(\frac{xe^{2x}}{2} - \frac{e^{2x}}{4}\right)\right]$ | A1 cao | |
| Substitutes limits $3$ and $1$ and subtracts | dM1 | Dependent on second and third Ms |
| $= \pi\left[\frac{13}{4}e^6 - \frac{1}{4}e^2\right]$ or any correct exact equivalent | A1 | **[8]** |
| Omission of $\pi$ loses first and last marks only | | |
---4.
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{9bf05d7e-7bb9-40f6-b626-69a8a6eda5a5-05_556_723_299_632}
\end{center}
\end{figure}
Figure 1 shows the finite shaded region, $R$, which is bounded by the curve $y = x \mathrm { e } ^ { x }$, the line $x = 1$, the line $x = 3$ and the $x$-axis.
The region $R$ is rotated through 360 degrees about the $x$-axis.\\
Use integration by parts to find an exact value for the volume of the solid generated.\\
(8)\\
\hfill \mbox{\textit{Edexcel C4 2006 Q4 [8]}}