A-LevelFurther MathematicsIntegration9231/12 · May/June 2019 · Q58 Marks

A curve C is defined parametrically by x = 2/(eᵗ + e⁻ᵗ) and y = (eᵗ - e⁻ᵗ)/(eᵗ + e⁻ᵗ), for 0 ≤ t ≤ 1. The area of the surface generated when C is rotated through 2π radians about the x-axis is denoted by S.

(i)Show that S = 4π ∫0^1 ((eᵗ - e⁻ᵗ)/(eᵗ + e⁻ᵗ)²) dt.[5]
(ii)Using the substitution u = eᵗ + e⁻ᵗ, or otherwise, find S in terms of π and e.[3]

Mark scheme

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

[2] Differentiates and simplifies. Accept 1 - ( (e^t - e^-t) / (e^t + e^-t) )^2.
[2] Attempt at writing (dx/dt)² + (dy/dt)² as a square.
[1] Uses correct formula, simplifies to AG.
[2] Applies given substitution.
[1] AEF, must be exact.

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Topic: Integration, A-Level Further Mathematics (syllabus 9231). Structured question, 8 marks.
Source: A-Level Further Mathematics, May/June 2019 Paper 1 Variant 2, question 5.

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