5 Kai uses the midpoint rule, trapezium rule and Simpson's rule to find approximations to \(\int _ { \mathrm { a } } ^ { \mathrm { b } } \mathrm { f } ( \mathrm { x } ) \mathrm { dx }\), where \(a\) and \(b\) are constants. The associated spreadsheet output is shown in the table. Some of the values are missing.
| F | G | H | I | |
| 3 | \(n\) | \(\mathrm { M } _ { \mathrm { n } }\) | \(\mathrm { T } _ { \mathrm { n } }\) | \(\mathrm { S } _ { 2 \mathrm { n } }\) | |
| 4 | 1 | 0.2436699 | 0.1479020 | | |
| 5 | 2 | 0.2306967 | | | |
- Write down a suitable spreadsheet formula for cell H 5 .
- Complete the copy of the table in the Printed Answer Booklet, giving the values correct to 7 decimal places.
- Use your answers to part (b) to determine the value of \(\int _ { a } ^ { b } f ( x ) d x\) as accurately as you can, justifying the precision quoted.