7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{96e004d9-c6b6-474b-9b67-06e1771c609e-14_604_1063_251_502}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{figure}
Figure 4 shows a sketch of part of the curve with equation
$$y = 2 \mathrm { e } ^ { 2 x } - x \mathrm { e } ^ { 2 x } , \quad x \in \mathbb { R }$$
The finite region \(R\), shown shaded in Figure 4, is bounded by the curve, the \(x\)-axis and the \(y\)-axis.
Use calculus to show that the exact area of \(R\) can be written in the form \(p \mathrm { e } ^ { 4 } + q\), where \(p\) and \(q\) are rational constants to be found.
(Solutions based entirely on graphical or numerical methods are not acceptable.)