OCR MEI C4 — Question 4 4 marks

Exam BoardOCR MEI
ModuleC4 (Core Mathematics 4)
Marks4
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicNumerical integration
TypeComplete table then apply trapezium rule
DifficultyModerate -0.3 This is a straightforward application of the trapezium rule requiring only calculator work to complete the table (evaluating √(cos x) at three points) and then applying the standard trapezium rule formula. Part (ii) tests basic understanding of trapezium rule over/under-estimation for concave curves. While it involves multiple steps, each is routine and requires no problem-solving insight—slightly easier than average due to its procedural nature.
Spec1.09f Trapezium rule: numerical integration
4
  1. Complete the table of values for the curve \(y = \sqrt { \cos x }\).
    \(X\)0\(\frac { \pi } { 8 }\)\(\frac { \pi } { 4 }\)\(\frac { 3 \pi } { 8 }\)\(\frac { \pi } { 2 }\)
    \(y\)0.96120.8409
    Hence use the trapezium rule with strip width \(h = \frac { \pi } { 8 }\) to estimate the value of the integral \(\int _ { 0 } ^ { \frac { \pi } { 2 } } \sqrt { \cos x } \mathrm {~d} x\), giving your answer to 3 decimal places. Fig. 4 shows the curve \(y = \sqrt { \cos x }\) for \(0 \leqslant x \leqslant \frac { \pi } { 2 }\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{ce44db53-2ec8-497b-a1d5-a8adf85e3929-4_459_751_799_638} \captionsetup{labelformat=empty} \caption{Fig. 4}
    \end{figure}
  2. 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.
Question 4:
Part (i)
AnswerMarks Guidance
AnswerMarks Guidance
\(1, 0.6186, 0\)B1 4dp (or more)
\(A \approx \frac{\pi}{16}\{1 + 0 + 2(0.9612 + 0.8409 + 0.6186)\}\)M1 ft their table; need to see trapezium rule
\(= 1.147\) (3 dp)A1 [3] cao
Part (ii)
AnswerMarks Guidance
AnswerMarks Guidance
The estimate will increase, because the trapezia will be below but closer to the curve, reducing the error.B1 [1] o.e., or an illustration using the curve; full answer required 
# Question 4:

## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $1, 0.6186, 0$ | B1 | 4dp (or more) |
| $A \approx \frac{\pi}{16}\{1 + 0 + 2(0.9612 + 0.8409 + 0.6186)\}$ | M1 | ft their table; need to see trapezium rule |
| $= 1.147$ (3 dp) | A1 [3] | cao |

## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| The estimate will increase, because the trapezia will be below but closer to the curve, reducing the error. | B1 [1] | o.e., or an illustration using the curve; full answer required |

---
4 (i) 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 & $\frac { \pi } { 8 }$ & $\frac { \pi } { 4 }$ & $\frac { 3 \pi } { 8 }$ & $\frac { \pi } { 2 }$ \\
\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 $\int _ { 0 } ^ { \frac { \pi } { 2 } } \sqrt { \cos x } \mathrm {~d} x$, giving your answer to 3 decimal places.

Fig. 4 shows the curve $y = \sqrt { \cos x }$ for $0 \leqslant x \leqslant \frac { \pi } { 2 }$.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{ce44db53-2ec8-497b-a1d5-a8adf85e3929-4_459_751_799_638}
\captionsetup{labelformat=empty}
\caption{Fig. 4}
\end{center}
\end{figure}

(ii) 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.

\hfill \mbox{\textit{OCR MEI C4  Q4 [4]}}
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Q1 5 Q2 8 Q3 9 Q4 4 Q5 6 Q6 8