18 5. (i) Evaluate the improper integral 2 1 1 2 e d ∞ − ∫ x x (3) (ii) The air temperature, θ °C, on a particular day in London is modelled by the equation θ = 8 – 5 sin  12 π t       – cos  6 π t       0  t  24 where t is the number of hours after midnight. (a) Use calculus to show that the mean air temperature on this day is 8 °C, according to the model. (3) Given that the actual mean air temperature recorded on this day was higher than 8 °C, (b) explain how the model could be refined. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________