| 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 -0.3 This is a straightforward C2 trapezium rule application with 4 intervals requiring routine substitution into the formula, followed by a standard concavity reasoning question. The function evaluation is simple and the over/under-estimate determination follows directly from observing the curve's shape, making this slightly easier than average. |
| Spec | 1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | x | 0 |
| \(\frac{4x}{(x+1)^2}\) | 0 | 0.64 |
| area \(= \frac{1}{8} \times 0.25 \times [0 + 1 + 2(0.64 + 0.8889 + 0.9796)]\) | B1 M1 | |
| \(= 0.752\) (3sf) | A1 | |
| (b) under-estimate | B1 | |
| the curve passes above the top edge of each trapezium | B1 | (7 marks) |
(a) | x | 0 | 0.25 | 0.5 | 0.75 | 1 |
| $\frac{4x}{(x+1)^2}$ | 0 | 0.64 | 0.8889 | 0.9796 | 1 | M1 A1 |
area $= \frac{1}{8} \times 0.25 \times [0 + 1 + 2(0.64 + 0.8889 + 0.9796)]$ | B1 M1 |
$= 0.752$ (3sf) | A1 |
(b) under-estimate | B1 |
the curve passes above the top edge of each trapezium | B1 | (7 marks)3.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{824da843-3ea1-4a83-9170-d0082b2e8d1c-2_476_880_1254_539}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows the curve with equation $y = \frac { 4 x } { ( x + 1 ) ^ { 2 } }$.\\
The shaded region is bounded by the curve, the $x$-axis and the line $x = 1$.
\begin{enumerate}[label=(\alph*)]
\item Use the trapezium rule with four intervals of equal width to find an estimate for the area of the shaded region.
\item State, with a reason, whether your answer to part (a) is an under-estimate or an over-estimate of the true area.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q3 [7]}}