- Evaluate the improper integral
$$\int _ { 1 } ^ { \infty } 2 \mathrm { e } ^ { - \frac { 1 } { 2 } x } \mathrm {~d} x$$
(ii) The air temperature, \(\theta ^ { \circ } \mathrm { C }\), on a particular day in London is modelled by the equation
$$\theta = 8 - 5 \sin \left( \frac { \pi } { 12 } t \right) - \cos \left( \frac { \pi } { 6 } t \right) \quad 0 \leqslant t \leqslant 24$$
where \(t\) is the number of hours after midnight.
- Use calculus to show that the mean air temperature on this day is \(8 ^ { \circ } \mathrm { C }\), according to the model.
Given that the actual mean air temperature recorded on this day was higher than \(8 ^ { \circ } \mathrm { C }\),
- explain how the model could be refined.