A-LevelFurther MathematicsIntegration9231/12 · May/June 2016 · Q1114 Marks

ANSWER ONLY ONE OF THE FOLLOWING TWO ALTERNATIVES. EITHER A curve C has parametric equations x = e²ᵗ cos 2t, y = e²ᵗ sin 2t, for -π/4 ≤ t ≤ π/4. Find the arc length of C. Find the area of the surface generated when C is rotated through 2π radians about the x-axis.

Find the arc length of C.[6]
Find the area of the surface generated when C is rotated through 2π radians about the x-axis.[8]

Mark scheme

Marking points from the official mark scheme (May/June 2016). The mark or mark type is shown in brackets.

[B1] x' = 2e²ᵗ cos 2t - 2e²ᵗ sin 2t, y' = 2e²ᵗ cos 2t + 2e²ᵗ sin 2t
[M1A1] x'² + y'² = 8e⁴ᵗ
[M1A1] s = ∫ (-π/4 to π/4) 2√2e²ᵗ dt = √2(e^(π/2) - e^(-π/2)) or 2√2 sinh(π/2) or 32.7
[M1] S = 2π ∫ (-π/4 to π/4) e²ᵗ sin 2t . √8 e²ᵗ dt = 4√2π ∫ (-π/4 to π/4) e⁴ᵗ sin 2t dt
[M1A1] Let I = ∫e⁴ᵗ sin 2t dt = [-e⁴ᵗ cos 2t / 2] + ∫2e⁴ᵗ cos 2t dt
[A1A1] I = (e^π - e^(-π))/10
[A1A1] S = 2√2π (e^π - e^(-π))/5 or 4√2π/5 sinh(π) or 952 (3sf)

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