5.
\begin{figure}[h]
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\caption{Figure 1}
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Figure 1 shows part of the curve with equation \(x = 4 t \mathrm { e } ^ { - \frac { 1 } { 3 } t } + 3\). The finite region \(R\) shown shaded in Figure 1 is bounded by the curve, the \(x\)-axis, the \(t\)-axis and the line \(t = 8\).
- Complete the table with the value of \(x\) corresponding to \(t = 6\), giving your answer to 3 decimal places.
| \(t\) | 0 | 2 | 4 | 6 | 8 |
| \(x\) | 3 | 7.107 | 7.218 | | 5.223 |
- Use the trapezium rule with all the values of \(x\) in the completed table to obtain an estimate for the area of the region \(R\), giving your answer to 2 decimal places.
- Use calculus to find the exact value for the area of \(R\).
- Find the difference between the values obtained in part (b) and part (c), giving your answer to 2 decimal places.