| Exam Board | Edexcel |
|---|---|
| Module | P4 (Pure Mathematics 4) |
| Year | 2021 |
| Session | October |
| Marks | 7 |
| 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 multi-step P4 question requiring integration by parts for part (a), then applying it to a volume of revolution in part (b). While the techniques are standard A-level material, the combination of integration by parts with x²ln(x), followed by setting up and evaluating the volume integral with limits involving e, requires careful algebraic manipulation and is more demanding than typical C3/C4 questions. The 'show all working' requirement and exact answer format add to the challenge. |
| Spec | 1.08i Integration by parts4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\int x^2 \ln x \, dx = \frac{1}{3}x^3 \ln x - \int \frac{1}{3}x^3 \times \frac{1}{x} \, dx\) | M1 | Attempts IBP the correct way around. Look for \(...x^3 \ln x - \int ...x^3 \times \frac{1}{x} \, dx\) |
| \(= \frac{1}{3}x^3 \ln x - \frac{1}{9}x^3 (+c)\) | dM1 A1 | Must reach form \(px^3 \ln x - qx^3\) where \(p,q\) positive. Condone missing \(+c\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\int x^2 \ln^2 x \, dx = \frac{1}{3}x^3 \ln^2 x - \int \frac{1}{3}x^3 \times 2\ln x \times \frac{1}{x} \, dx\) | M1 | IBP correct way to reach form \(...x^3\ln^2 x - ...\int x^3 \times \ln x \times \frac{1}{x} \, dx\) |
| \(= \frac{1}{3}x^3\ln^2 x - \frac{2}{3}\left(\frac{1}{3}x^3 \ln x - \frac{1}{9}x^3\right)\) | A1ft | Achieves \(\frac{1}{3}x^3\ln^2 x - \frac{2}{3} \times\) their answer to (a) |
| \(\text{Volume} = \int_1^e \pi x^2 \ln^2 x \, dx = \pi \times \left[\frac{1}{3}x^3\ln^2 x - \frac{2}{3}\left(\frac{1}{3}x^3\ln x - \frac{1}{9}x^3\right)\right]_1^e\) | dM1 | Applies limits 1 and e to function of form \(\pi\times\left[\alpha x^3\ln^2 x - \beta x^3 \ln x \pm \chi x^3\right]_1^e\), \(\pi\) must be seen |
| \(= \frac{5}{27}\pi e^3 - \frac{2}{27}\pi\) | A1 | Or exact equivalent e.g. \(\frac{\pi}{27}(5e^3-2)\) |
# Question 8:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int x^2 \ln x \, dx = \frac{1}{3}x^3 \ln x - \int \frac{1}{3}x^3 \times \frac{1}{x} \, dx$ | M1 | Attempts IBP the correct way around. Look for $...x^3 \ln x - \int ...x^3 \times \frac{1}{x} \, dx$ |
| $= \frac{1}{3}x^3 \ln x - \frac{1}{9}x^3 (+c)$ | dM1 A1 | Must reach form $px^3 \ln x - qx^3$ where $p,q$ positive. Condone missing $+c$ |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int x^2 \ln^2 x \, dx = \frac{1}{3}x^3 \ln^2 x - \int \frac{1}{3}x^3 \times 2\ln x \times \frac{1}{x} \, dx$ | M1 | IBP correct way to reach form $...x^3\ln^2 x - ...\int x^3 \times \ln x \times \frac{1}{x} \, dx$ |
| $= \frac{1}{3}x^3\ln^2 x - \frac{2}{3}\left(\frac{1}{3}x^3 \ln x - \frac{1}{9}x^3\right)$ | A1ft | Achieves $\frac{1}{3}x^3\ln^2 x - \frac{2}{3} \times$ their answer to (a) |
| $\text{Volume} = \int_1^e \pi x^2 \ln^2 x \, dx = \pi \times \left[\frac{1}{3}x^3\ln^2 x - \frac{2}{3}\left(\frac{1}{3}x^3\ln x - \frac{1}{9}x^3\right)\right]_1^e$ | dM1 | Applies limits 1 and e to function of form $\pi\times\left[\alpha x^3\ln^2 x - \beta x^3 \ln x \pm \chi x^3\right]_1^e$, $\pi$ must be seen |
| $= \frac{5}{27}\pi e^3 - \frac{2}{27}\pi$ | A1 | Or exact equivalent e.g. $\frac{\pi}{27}(5e^3-2)$ |
---\begin{enumerate}[label=(\alph*)]
\item Find $\int x ^ { 2 } \ln x \mathrm {~d} x$
Figure 3 shows a sketch of part of the curve with equation
$$y = x \ln x \quad x > 0$$
The region $R$, shown shaded in Figure 3, lies entirely above the $x$-axis and is bounded by the curve, the $x$-axis and the line with equation $x = \mathrm { e }$.
This region is rotated through $2 \pi$ radians about the $x$-axis to form a solid of revolution.
\item Find the exact volume of the solid formed, giving your answer in simplest form.
\section*{8. In this question you must show all stages of your working. \\
In this question you must show all stages of your working.}
\end{enumerate}
\hfill \mbox{\textit{Edexcel P4 2021 Q8 [7]}}