Question 4 - A-Level Maths

Exam Board	OCR MEI
Module	Further Numerical Methods (Further Numerical Methods)
Year	2020
Session	November
Marks	4
Topic	Sign Change & Interval Methods
Type	Trapezium Rule with Accuracy Analysis

Use the trapezium rule with 1 strip to calculate an estimate of \(\int _ { 1 } ^ { 2 } \sqrt { 1 + x ^ { 3 } } \mathrm {~d} x\), giving your
answer correct to six decimal places.
[0pt] [2]
Fig. 4 shows some spreadsheet output containing further approximations to this integral using the trapezium rule, denoted by \(T _ { n }\), and Simpson's rule, denoted by \(S _ { 2 n }\). \begin{table}[h]
A B C
1 \(n\) \(T _ { n }\) \(S _ { 2 n }\)
2 1 2.130135
3 2 2.149378
4 4 2.134751 2.129862
5 8 2.131084
\captionsetup{labelformat=empty} \caption{Fig. 4}
\end{table}
Write down an efficient formula for cell C 4.
Find the value of \(S _ { 4 }\), giving your answer correct to 6 decimal places.
Without doing any further calculation, state the value of \(\int _ { 1 } ^ { 2 } \sqrt { 1 + x ^ { 3 } } \mathrm {~d} x\) as accurately as
possible, justifying the precision quoted.
[0pt] [2]
Use the fact that Simpson's rule is a fourth order method to obtain an improved approximation to the value of \(\int _ { 1 } ^ { 2 } \sqrt { 1 + x ^ { 3 } } \mathrm {~d} x\), stating the value of this integral to a precision which seems justified.