A-LevelMathematicsIntegration9709/32 · Feb/Mar 2019 · Q1012 Marks

The diagram shows the curve y = sin³x √(cos x) for 0 ≤ x ≤ ½π, and its maximum point M. [Figure 10.1]

(i)Using the substitution u = cos x, find by integration the exact area of the shaded region bounded by the curve and the x-axis.[6]
(ii)Showing all your working, find the x-coordinate of M, giving your answer correct to 3 decimal places.[6]

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Mark scheme

Marking points from the official mark scheme (Feb/Mar 2019). The mark or mark type is shown in brackets.

[B1] State or imply du = – sin x dx
[M1] Using Pythagoras express the integral in terms of u
[A1] Obtain integrand u^(1/2) (1-u²) or u^(1/2) - u^(5/2), or equivalent
[A1] Integrate and obtain 2/3 u^(3/2) - 2/7 u^(7/2), or equivalent
[M1] Change limits correctly and substitute correctly in an integral of the form au^(3/2) + bu^(7/2) Or substitute original limits correctly in an integral of the form a(cos x)^(3/2) + b(cos x)^(7/2)
[A1] Obtain answer 8/21
[M1] Use product rule and chain rule at least once
[A1 + A1] Obtain correct derivative in any form
[M1] Equate derivative to zero and obtain a horizontal equation in integral powers of sin x and cos x
[M1] Use correct methods to obtain an equation in one trig function
[A1] Obtain tan²x = 6, 7cos²x=1 or 7sin²x = 6, or equivalent, and obtain answer 1.183

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