A-LevelMathematicsIntegration9709/21 · May/June 2024 · Q69 Marks
The diagram shows the curve with equation y = √sin2x+sin²2x for 0 < x < π. The shaded region is bounded by the curve and the straight lines x = π and y = 0. [Figure X.X]
- (a)Use the trapezium rule with two intervals to find an approximation to the area of the shaded region. Give your answer correct to 2 significant figures.[3]
- (b)The shaded region is rotated completely about the x-axis. Find the exact volume of the solid produced.[6]
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Mark scheme
Marking points from the official mark scheme (May/June 2024). The mark or mark type is shown in brackets.
- [B1] Use y-values (0), √sin(π/6)+sin²(π/6), √sin(π/3)+sin²(π/3) or decimal equivalents
- [M1] Use correct formula, or equivalent, with h = π/12
- [A1] Obtain 0.39
- [M1] Use (π)∫(sin 2x + sin² 2x) dx
- [*M1] Express integrand in the form k₁ sin 2x + k₂ + k₃ cos 4x
- [A1] Obtain correct sin 2x + 1/2 - 1/2 cos 4x
- [A1] Integrate to obtain -1/2 cos 2x + 1/2 x - 1/8 sin 4x
- [DM1] Apply limits correctly to obtain exact value
- [A1] Obtain volume = π/4 + π²/12 - π√3/16
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