Given that \(\mathrm { e } ^ { - 2 x } = 4\), find the exact value of \(x\).
The diagram shows the curve \(y = 4 \mathrm { e } ^ { - 2 x } - \mathrm { e } ^ { - 4 x }\).
\includegraphics[max width=\textwidth, alt={}, center]{6761e676-48ae-47e9-9617-153342cdf5c4-9_490_1185_463_440}
The curve crosses the \(y\)-axis at the point \(A\), the \(x\)-axis at the point \(B\), and has a stationary point at \(M\).
State the \(y\)-coordinate of \(A\).
Find the \(x\)-coordinate of \(B\), giving your answer in an exact form.
Find the \(x\)-coordinate of the stationary point, \(M\), giving your answer in an exact form.
The shaded region \(R\) is bounded by the curve \(y = 4 \mathrm { e } ^ { - 2 x } - \mathrm { e } ^ { - 4 x }\), the lines \(x = 0\) and \(x = \ln 2\) and the \(x\)-axis.
Find the volume of the solid generated when the region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis, giving your answer in the form \(\frac { p } { q } \pi\), where \(p\) and \(q\) are integers.
(7 marks)