| Date | May 2014 | Marks available | 3 | Reference code | 14M.2.hl.TZ2.3 |
| Level | HL only | Paper | 2 | Time zone | TZ2 |
| Command term | Find | Question number | 3 | Adapted from | N/A |
Question
The graphs of \(y = {x^2}{{\text{e}}^{ - x}}\) and \(y = 1 - 2\sin x\) for \(2 \leqslant x \leqslant 7\) intersect at points A and B.
The x-coordinates of A and B are \({x_{\text{A}}}\) and \({x_{\text{B}}}\).
Find the value of \({x_{\text{A}}}\) and the value of \({x_{\text{B}}}\).
[2]
a.
Find the area enclosed between the two graphs for \({x_{\mathbf{A}}} \leqslant x \leqslant {x_{\text{B}}}\).
[3]
b.
Markscheme
\({x_{\text{A}}} = 2.87\) A1
\({x_{{\text{B}}}} = 6.78\) A1
[2 marks]
a.
\(\int_{2.87172{\text{K}}}^{6.77681K} {1 - 2\sin x - {x^2}{{\text{e}}^{ - x}}{\text{d}}x} \) (M1)(A1)
\( = 6.76\) A1
Note: Award (M1) for definite integral and (A1) for a correct definite integral.
[3 marks]
b.
Examiners report
[N/A]
a.
[N/A]
b.
Syllabus sections
Topic 6 - Core: Calculus » 6.5 » Area of the region enclosed by a curve and the \(x\)-axis or \(y\)-axis in a given interval; areas of regions enclosed by curves.
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