7. The curve \(C\) has equation
$$y = \mathrm { e } ^ { - x } , \quad x \in \mathbb { R }$$
The part of the curve \(C\) between \(x = 0\) and \(x = \ln 3\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
- Show that the area \(S\) of the curved surface generated is given by
$$S = 2 \pi \int _ { 0 } ^ { \ln 3 } \mathrm { e } ^ { - x } \sqrt { 1 + \mathrm { e } ^ { - 2 x } } \mathrm {~d} x$$
- Use the substitution \(\mathrm { e } ^ { - x } = \sinh u\) to show that
$$S = 2 \pi \int _ { \operatorname { arsinh } \alpha } ^ { \operatorname { arsinh } \beta } \cosh ^ { 2 } u \mathrm {~d} u$$
where \(\alpha\) and \(\beta\) are constants to be determined.
- Show that
$$2 \int \cosh ^ { 2 } u \mathrm {~d} u = \frac { 1 } { 2 } \sinh 2 u + u + k$$
where \(k\) is an arbitrary constant.
- Hence find the value of \(S\), giving your answer to 3 decimal places.