2.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e14881c1-5ba5-4868-92ee-8bc58d4884dc-03_606_1070_251_445}
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\caption{Figure 1}
\end{figure}
Figure 1 shows a sketch of part of the curve with equation
$$y = ( 2 - x ) \mathrm { e } ^ { 2 x } , \quad x \in \mathbb { R }$$
The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the \(y\)-axis.
The table below shows corresponding values of \(x\) and \(y\) for \(y = ( 2 - x ) \mathrm { e } ^ { 2 x }\)
| \(x\) | 0 | 0.5 | 1 | 1.5 | 2 |
| \(y\) | 2 | 4.077 | 7.389 | 10.043 | 0 |
- Use the trapezium rule with all the values of \(y\) in the table, to obtain an approximation for the area of \(R\), giving your answer to 2 decimal places.
- Explain how the trapezium rule can be used to give a more accurate approximation for the area of \(R\).
- Use calculus, showing each step in your working, to obtain an exact value for the area of \(R\). Give your answer in its simplest form.