| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2012 |
| Session | January |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Complete table then apply trapezium rule |
| Difficulty | Moderate -0.3 This is a straightforward application of the trapezium rule with routine function evaluation. Part (i) requires calculating √(cos x) at three x-values and applying the standard trapezium rule formula—pure procedural work. Part (ii) tests basic understanding that the trapezium rule overestimates for concave curves, which is standard bookwork. The question is slightly easier than average because it's entirely mechanical with no problem-solving or conceptual challenge beyond recalling the method. |
| Spec | 1.09f Trapezium rule: numerical integration |
| \(x\) | 0 | \(\frac{\pi}{6}\) | \(\frac{\pi}{4}\) | \(\frac{3\pi}{8}\) | \(\frac{\pi}{2}\) |
| \(y\) | 0.9612 | 0.8409 |
Question 4:
--- 4 (i) ---
4 (i)
--- 4 (ii) ---
4 (ii)\begin{enumerate}[label=(\roman*)]
\item Complete the table of values for the curve $y = \sqrt{\cos x}$.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$x$ & 0 & $\frac{\pi}{6}$ & $\frac{\pi}{4}$ & $\frac{3\pi}{8}$ & $\frac{\pi}{2}$ \\
\hline
$y$ & & 0.9612 & 0.8409 & & \\
\hline
\end{tabular}
\end{center}
Hence use the trapezium rule with strip width $h = \frac{\pi}{8}$ to estimate the value of the integral $\int_0^{\frac{\pi}{2}} \sqrt{\cos x} \, dx$, giving your answer to 3 decimal places. [3]
Fig. 4 shows the curve $y = \sqrt{\cos x}$ for $0 \leq x \leq \frac{\pi}{2}$.
\includegraphics{figure_4}
\item State, with a reason, whether the trapezium rule with a strip width of $\frac{\pi}{16}$ would give a larger or smaller estimate of the integral. [1]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C4 2012 Q4 [4]}}