The curve C has parametric equations x = t², y = t - ⅓ t³, for 0 ≤ t ≤ 1. Find (i) the arc length of C, (ii) the surface area generated when C is rotated through 2π radians about the x-axis.

Mark scheme

Marking points from the official mark scheme (May/June 2014). The mark or mark type is shown in brackets.

1. [B1] dx/dt = 2t, dy/dt = 1 - t²
2. [M1A1] (dx/dt)² + (dy/dt)² = 4t² + (1 - t²)² = 4t² + 1 - 2t² + t⁴ = (1 + t²)² ACF
3. [M1A1] s = ∫(from 0 to 1) √((dx/dt)² + (dy/dt)²) dt = ∫(from 0 to 1) (1 + t²) dt = [t + t³/3] (from 0 to 1) or 1.33 LR
4. [M1A1] S = 2π ∫(from 0 to 1) y √((dx/dt)² + (dy/dt)²) dt = 2π ∫(from 0 to 1) (t - t³/3)(1 + t²) dt LNR
5. [M1A1] = 2π ∫(from 0 to 1) (t + t³ - t³/3 - t⁵/3) dt = 2π ∫(from 0 to 1) (t + (2/3)t³ - t⁵/3) dt LR
6. [A1] = 2π [t²/2 + (2/3)(t⁴/4) - (1/3)(t⁶/6)] (from 0 to 1)
7. [A1] = 2π [½ + ⅙ - ⅛] = 11π/9 or 3.84