| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Trapezium rule then exact integration comparison |
| Difficulty | Moderate -0.8 This is a straightforward C4 question testing standard trapezium rule application and basic integration. Part (a) requires simple substitution, (b) is routine trapezium rule formula application, (c) involves standard integration of 3/x and x^4, and (d) is a simple percentage calculation. All steps are mechanical with no problem-solving or insight required, making it easier than average. |
| Spec | 1.08d Evaluate definite integrals: between limits1.09f Trapezium rule: numerical integration |
| \(x\) | 0.5 | 0.75 | 1 | 1.25 | 1.5 |
| \(\frac{3}{x} + x^4\) | 6.06 | 4.32 |
A student tests the accuracy of the trapezium rule by evaluating $I$, where
$$I = \int_{0.5}^{1.5} \left(\frac{3}{x} + x^4\right) dx.$$
\begin{enumerate}[label=(\alph*)]
\item Complete the student's table, giving values to 2 decimal places where appropriate.
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$x$ & 0.5 & 0.75 & 1 & 1.25 & 1.5 \\
\hline
$\frac{3}{x} + x^4$ & 6.06 & 4.32 & & & \\
\hline
\end{tabular}
[2]
\item Use the trapezium rule, with all the values from your table, to calculate an estimate for the value of $I$. [4]
\item Use integration to calculate the exact value of $I$. [4]
\item Verify that the answer obtained by the trapezium rule is within 3\% of the exact value. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 Q3 [12]}}