5.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{5a77103e-73c1-4f32-af93-f2d5627e2290-02_467_783_317_303}
\end{figure}
Figure 1 shows the graph of the curve with equation
$$y = x \mathrm { e } ^ { 2 x } , \quad x \geq 0 .$$
The finite region \(R\) bounded by the lines \(x = 1\), the \(x\)-axis and the curve is shown shaded in Figure 1.
- Use integration to find the exact value of the area for \(R\).
- Complete the table with the values of \(y\) corresponding to \(x = 0.4\) and 0.8 .
| \(x\) | 0 | 0.2 | 0.4 | 0.6 | 0.8 | 1 |
| \(y = x \mathrm { e } ^ { 2 x }\) | 0 | 0.29836 | | 1.99207 | | 7.38906 |
- Use the trapezium rule with all the values in the table to find an approximate value for this area, giving your answer to 4 significant figures.