9
\includegraphics[max width=\textwidth, alt={}, center]{71e01d8f-d0ed-4f17-b7cd-6f5a93bbe329-4_661_915_269_557}
The diagram shows the curve
$$y = \mathrm { e } ^ { 2 x } - 18 x + 15 .$$
The curve crosses the \(y\)-axis at \(P\) and the minimum point is \(Q\). The shaded region is bounded by the curve and the line \(P Q\).
- Show that the \(x\)-coordinate of \(Q\) is \(\ln 3\).
- Find the exact area of the shaded region.