Given that \(\mathrm { e } ^ { - 2 x } = 3\), find the exact value of \(x\).
Use integration by parts to find \(\int x \mathrm { e } ^ { - 2 x } \mathrm {~d} x\).
A curve has equation \(y = \mathrm { e } ^ { - 2 x } + 6 x\).
Find the exact values of the coordinates of the stationary point of the curve.
Determine the nature of the stationary point.
The region \(R\) is bounded by the curve \(y = \mathrm { e } ^ { - 2 x } + 6 x\), the \(x\)-axis and the lines \(x = 0\) and \(x = 1\).
Find the volume of the solid formed when \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis, giving your answer to three significant figures.