| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Multi-part: volume and area |
| Difficulty | Standard +0.3 This is a straightforward C4 volumes of revolution question with standard techniques: trapezium rule application (routine calculation with given intervals) and volume integration using a substitution or integration by parts. The function is slightly more complex than basic polynomials due to the exponential, but the methods are standard textbook exercises requiring no novel insight. |
| Spec | 1.09f Trapezium rule: numerical integration4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | \(x\) | 0 |
| \(y\) | 0 | 1.716 |
| Area \(= \frac{1}{2} \times 0.5 \times [0 + 1.966 + 2(1.716 + 1.472 + 1.093)] = \frac{2.33}{0.766}\) (3sf) | B2 B1 M1 A1 | |
| (b) Volume \(= \pi \int_0^2 16x e^{-2x} dx\) | M1 | |
| \(u = 16x\), \(u' = 16\), \(v' = e^{-2x}\), \(v = -\frac{1}{2}e^{-2x}\) | M1 | |
| \(I = -8x e^{-2x} - \int -8e^{-2x} dx\) | A2 | |
| \(= -8x e^{-2x} - 4e^{-2x} + c\) | A1 | |
| Volume \(= \pi[-8x e^{-2x} - 4e^{-2x}]_0^2\) | M1 | |
| \(= \pi[(-16e^{-4} - 4e^{-4}) - (0 - 4)]\) | M1 | |
| \(= 4\pi(1 - 5e^{-4})\) | A1 | (12 marks) |
**(a)** | $x$ | 0 | 0.5 | 1 | 1.5 | 2 |
|---|---|---|---|---|---|
| $y$ | 0 | 1.716 | 1.472 | 1.093 | 0.966 |
Area $= \frac{1}{2} \times 0.5 \times [0 + 1.966 + 2(1.716 + 1.472 + 1.093)] = \frac{2.33}{0.766}$ (3sf) | B2 B1 M1 A1 |
**(b)** Volume $= \pi \int_0^2 16x e^{-2x} dx$ | M1 |
$u = 16x$, $u' = 16$, $v' = e^{-2x}$, $v = -\frac{1}{2}e^{-2x}$ | M1 |
$I = -8x e^{-2x} - \int -8e^{-2x} dx$ | A2 |
$= -8x e^{-2x} - 4e^{-2x} + c$ | A1 |
Volume $= \pi[-8x e^{-2x} - 4e^{-2x}]_0^2$ | M1 |
$= \pi[(-16e^{-4} - 4e^{-4}) - (0 - 4)]$ | M1 |
$= 4\pi(1 - 5e^{-4})$ | A1 | (12 marks)
---5.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{e877dc80-4cfc-4c8b-9640-9b186cd7ab13-08_617_917_146_475}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows the curve with equation $y = 4 x ^ { \frac { 1 } { 2 } } \mathrm { e } ^ { - x }$.\\
The shaded region is bounded by the curve, the $x$-axis and the line $x = 2$.
\begin{enumerate}[label=(\alph*)]
\item Use the trapezium rule with four intervals of equal width to estimate the area of the shaded region.
The shaded region is rotated through $2 \pi$ radians about the $x$-axis.
\item Find, in terms of $\pi$ and e, the exact volume of the solid formed.\\
5. continued
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 Q5 [12]}}