- A curve has parametric equations
$$x = 4 \mathrm { e } ^ { \frac { 1 } { 2 } t } \quad y = \mathrm { e } ^ { t } - t \quad 0 \leqslant t \leqslant 4$$
The curve is rotated through \(2 \pi\) radians about the \(x\)-axis.
Show that the area of the curved surface generated is
$$\pi \left( \mathrm { e } ^ { 8 } + A \mathrm { e } ^ { 4 } + B \right)$$
where \(A\) and \(B\) are constants to be determined.