7 A curve has equation \(y = 4 \sqrt { x }\).
- Show that the length of arc \(s\) of the curve between the points where \(x = 0\) and \(x = 1\) is given by
$$s = \int _ { 0 } ^ { 1 } \sqrt { \frac { x + 4 } { x } } \mathrm {~d} x$$
- Use the substitution \(x = 4 \sinh ^ { 2 } \theta\) to show that
$$\int \sqrt { \frac { x + 4 } { x } } \mathrm {~d} x = \int 8 \cosh ^ { 2 } \theta \mathrm {~d} \theta$$
- Hence show that
$$s = 4 \sinh ^ { - 1 } 0.5 + \sqrt { 5 }$$