| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Trapezium rule with stated number of strips |
| Difficulty | Moderate -0.5 This is a straightforward application of the trapezium rule with clearly specified strips (n=4), followed by a simple volume calculation using scaling. The function is given, no algebraic manipulation is needed, and the method is entirely procedural with no problem-solving insight required. Slightly easier than average due to its routine nature, though the applied context in part (b) adds minimal complexity. |
| Spec | 1.02z Models in context: use functions in modelling1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| \(x\) | 0 | 0.5 |
| \(\frac{1}{x^2+1}\) | 1 | 0.8 |
| M1 A1 | ||
| Area \(= \frac{1}{2} \times 0.5 \times [1 + 0.2 + 2(0.8 + 0.5 + 0.3077)]\) | B1 M1 | |
| \(= 1.10 \text{ (3sf)}\) | A1 |
| Answer | Marks |
|---|---|
| Area \(= 8^2 \times 1.10385 = 70.6464\) | M1 |
| Volume \(= 2 \times 70.6464 = 141 \text{ cm}^3 \text{ (3sf)}\) | A1 |
**(a)**
| $x$ | 0 | 0.5 | 1 | 1.5 | 2 |
|---|---|---|---|---|---|
| $\frac{1}{x^2+1}$ | 1 | 0.8 | 0.5 | 0.3077 | 0.2 |
| M1 A1 |
Area $= \frac{1}{2} \times 0.5 \times [1 + 0.2 + 2(0.8 + 0.5 + 0.3077)]$ | B1 M1 |
$= 1.10 \text{ (3sf)}$ | A1 |
**(b)**
Area $= 8^2 \times 1.10385 = 70.6464$ | M1 |
Volume $= 2 \times 70.6464 = 141 \text{ cm}^3 \text{ (3sf)}$ | A1 |3.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{089f5506-94ac-489f-b219-e67fa6ca834f-2_439_848_1560_461}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
Figure 2 shows the curve with equation $y = \frac { 1 } { x ^ { 2 } + 1 }$.\\
The shaded region $R$ is bounded by the curve, the coordinate axes and the line $x = 2$.
\begin{enumerate}[label=(\alph*)]
\item Use the trapezium rule with four strips of equal width to estimate the area of $R$.
The cross-section of a support for a bookshelf is modelled by $R$ with 1 unit on each axis representing 8 cm . Given that the support is 2 cm thick,
\item find an estimate for the volume of the support.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q3 [7]}}