2024 MI P1 Q3

A line \(l\) has equation \(y={{\mathrm{e}}^{4}}x\), \(x\ge 0\) and curve \(C\) has equation \(y=\sqrt{2x}{{\mathrm{e}}^{{{x}^{2}}}}\), \(x\ge 0\).

Both \(l\) and \(C\) intersect at the points with coordinates \(\left( 0,0 \right)\) and \(\left( 2,2{{\mathrm{e}}^{4}} \right)\) as shown in the diagram below.

1. The region \({R}\) is bounded by the line \({l}\), curve \({C}\), and the lines \({x = 1}\) and \({x = 3}\). Find, correct to \(4\) significant figures, the area of region \({R}\).[3]
2. The region \({S}\) is bounded by the line \({l}\) and curve \({C}\). Show that the volume \({V}\) of the solid formed when \({S}\) is rotated \({2\pi}\) radians about the \(x\)-axis is \({V = \frac{\pi}{6} \left( Ae^8 + B \right)}\), where \({A}\) and \({B}\) are exact constants to be determined.[4]