A-LevelMathematicsIntegration9709/31 · Oct/Nov 2018 · Q79 Marks

The diagram shows the curve y = 5 sin²x cos³x for 0 ≤ x ≤ ½π, and its maximum point M. The shaded region R is bounded by the curve and the x-axis. [Figure 7.1]

(i)Find the x-coordinate of M, giving your answer correct to 3 decimal places.[5]
(ii)Using the substitution u = sin x and showing all necessary working, find the exact area of R.[4]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2018). The mark or mark type is shown in brackets.

[M1*] (i) Use product rule
[A1] (i) Obtain correct derivative in any form
[depM1*] (i) Equate derivative to zero and obtain an equation in a single trig function
[A1] (i) Obtain a correct equation, e.g. 3 tan²x = 2
[A1] (i) Obtain answer x = 0.685
[M1] (ii) Use the given substitution and reach a ∫(u²-u⁴)du
[A1] (ii) Obtain correct integral with a = 5 and limits 0 and 1
[M1] (ii) Use correct limits in an integral of the form a(1/3 u³ - 1/5 u⁵)
[A1] (ii) Obtain answer 2/3

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