5
\includegraphics[max width=\textwidth, alt={}, center]{373fa8e4-9c10-4fcf-9e00-e497161b4c6d-05_339_869_262_244}
The diagram shows the curve \(C\) with parametric equations
\(x = \frac { 3 } { t } , y = t ^ { 3 } \mathrm { e } ^ { - 2 t }\), where \(t > 0\).
The maximum point on \(C\) is denoted by \(P\).
- Determine the exact coordinates of \(P\).
The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 6\).
- Show that the area of \(R\) is given by
$$\int _ { a } ^ { b } 3 t \mathrm { e } ^ { - 2 t } \mathrm {~d} t ,$$
where \(a\) and \(b\) are constants to be determined.
- Hence determine the exact area of \(R\).