A curve has equation \(y = x ^ { 2 } \mathrm { e } ^ { - \frac { x } { 4 } }\).
Show that the curve has exactly two stationary points and find the exact values of their coordinates.
(7 marks)
Use integration by parts twice to find the exact value of \(\int _ { 0 } ^ { 4 } x ^ { 2 } \mathrm { e } ^ { - \frac { x } { 4 } } \mathrm {~d} x\).
The region bounded by the curve \(y = 3 x \mathrm { e } ^ { - \frac { x } { 8 } }\), the \(x\)-axis from 0 to 4 and the line \(x = 4\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis to form a solid.
Use your answer to part (b)(i) to find the exact value of the volume of the solid generated.