Let \(f(x) = \cos ({x^2})\) and \(g(x) = {{\rm{e}}^x}\) , for \( - 1.5 \le x \le 0.5\) .

Find the area of the region enclosed by the graphs of f and g .

evidence of finding intersection points     (M1)

e.g. \(f(x) = g(x)\) , \(\cos {x^2} = {{\rm{e}}^x}\) , sketch showing intersection

\(x = - 1.11\) , \(x = 0\) (may be seen as limits in the integral)     A1A1

evidence of approach involving integration and subtraction (in any order)     (M1)

e.g. \(\int_{ - 1.11}^0 {\cos {x^2} - {{\rm{e}}^x}} \) , \(\int {(\cos {x^2} - {{\rm{e}}^x}){\rm{d}}x} \) , \(\int {g - f} \)

\({\text{area}} = 0.282\)     A2     N3

[6 marks]

This question was poorly done by a great many candidates. Most seemed not to understand what was meant by the phrase "region enclosed by" as several candidates assumed that the limits of the integral were those given in the domain. Few realized what area was required, or that intersection points were needed. Candidates who used their GDCs to first draw a suitable sketch could normally recognize the required region and could find the intersection points correctly. However, it was disappointing to see the number of candidates who could not then use their GDC to find the required area or who attempted unsuccessful analytical approaches.