A-LevelMathematicsIntegration9709/23 · Oct/Nov 2018 · Q68 Marks
The diagram shows the curve with equation y = √(1 + 3cos²(x/2)) for 0 ≤ x ≤ π. The region R is bounded by the curve, the axes and the line x = π. [Figure X.X]
- (i)Use the trapezium rule with two intervals to find an approximation to the area of R, giving your answer correct to 3 significant figures.[3]
- (ii)The region R is rotated completely about the x-axis. Without using a calculator, find the exact volume of the solid produced.[5]
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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.
- [B1] Use y values 2, √2.5, 1 or equivalents
- [M1] Use correct formula, or equivalent, with h = ½π and three y values
- [A1] Obtain ½π(2 + 2√2.5 + 1) or equivalent and hence 4.84
- [B1] State or imply volume is ∫₀^π π(1 + 3cos²(x/2)) dx
- [M1] Use appropriate identity to express integrand in form k₁ + k₂ cos x
- [A1] Obtain ∫π(5/2 + 3/2 cos(x)) dx or ∫(π(5/2 + 3/2 cos(x))) dx
- [A1] Integrate to obtain π(⁵/₂x + ³/₂sin(x)) or ⁵/₂πx + ³/₂πsin(x)
- [A1] Obtain ⁵/₂π² with no errors seen
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