9 The trapezium rule is used to calculate 3 approximations to \(\int _ { 0 } ^ { 1 } \sqrt [ 3 ] { \sinh ( x ) } \mathrm { d } x\) with 1,2 and 4 strips respectively. The results are shown in Table 9.1.
\begin{table}[h]
\captionsetup{labelformat=empty}
\caption{Table 9.1}
| \(n\) | \(\mathrm {~T} _ { n }\) |
| 1 | 0.52764369 |
| 2 | 0.66617652 |
| 4 | 0.72534275 |
\end{table}
- Use these results to determine two approximations to \(\int _ { 0 } ^ { 1 } \sqrt [ 3 ] { \sinh ( x ) } \mathrm { d } x\) using Simpson's rule.
- Use your answers to part (a) to state the value of \(\int _ { 0 } ^ { 1 } \sqrt [ 3 ] { \sinh ( x ) } \mathrm { d } x\) as accurately as you can, justifying the precision quoted.
Table 9.2 shows some further approximations found using the trapezium rule, together with some analysis of these approximations.
\begin{table}[h]
\captionsetup{labelformat=empty}
\caption{Table 9.2}
| \(n\) | \(\mathrm { T } _ { n }\) | difference | ratio |
| 1 | 0.5276437 | | |
| 2 | 0.6661765 | 0.138533 | |
| 4 | 0.7253427 | 0.059166 | 0.42709 |
| 8 | 0.7498821 | 0.024539 | 0.41475 |
| 16 | 0.7598858 | 0.010004 | 0.40766 |
| 32 | 0.7639221 | 0.004036 | 0.40348 |
| 64 | 0.7655404 | 0.001618 | 0.40095 |
- Explain what can be deduced about the order of the method in this case.
- Use extrapolation to obtain the value of \(\int _ { 0 } ^ { 1 } \sqrt [ 3 ] { \sinh ( x ) } \mathrm { d } x\) as accurately as you can, justifying the precision quoted.