6 The spreadsheet output in Fig. 6 shows approximations to \(\int _ { 0 } ^ { 1 } x ^ { - \sqrt { x } } \mathrm {~d} x\) found using the midpoint rule, denoted by \(M _ { n }\), and the trapezium rule, denoted by \(T _ { n }\).
\begin{table}[h]
| A | B | C | |
| 1 | \(n\) | \(M _ { n }\) | \(T _ { n }\) | |
| 2 | 1 | 1.632527 | 1 | |
| 3 | 2 | 1.641461 | 1.316263 | |
| 4 | 4 | 1.623053 | 1.478862 | |
| 5 | 8 | 1.610295 | 1.550957 | |
| 6 | 16 | 1.604132 | 1.580626 | |
| 7 | 32 | 1.601505 | 1.592379 | |
\captionsetup{labelformat=empty}
\caption{Fig. 6}
\end{table}
- Write down an efficient spreadsheet formula for cell C3.
- By first completing the table in the Printed Answer Booklet using the Simpson's rule, calculate the most accurate estimate of \(\int _ { 0 } ^ { 1 } x ^ { - \sqrt { x } } \mathrm {~d} x\) that you can, justifying the precision quoted.
\section*{END OF QUESTION PAPER}