The curve C has parametric equations x = t², y=t-t³, for 0 ≤ t ≤ 1. Find the surface area generated when C is rotated through 2π radians about the x-axis. Find the coordinates of the centroid of the region bounded by C, the x-axis and the line x = 1.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2013). The mark or mark type is shown in brackets.

1. [M1A1] (a) Obtains area of surface of revolution. ẋ = 2t, ẏ = 1-t²
2. [M1A1] ⇒ (ds/dt)² = 4t² + 1 - 2t² + t⁴ = (1+t²)²
3. [M1] 2π ∫ y ds = 2π ∫_0^1 (t - t³) (1+t²) dt
4. [A1] = 2π ∫_0^1 (t + t³ - t³ - t⁵) dt = 2π [t²/2 + t⁴/4 - t⁴/4 - t⁶/6]_0^1 = 11/9 π or 3.84
5. [B1] (b) Finds coordinates of centroid, using relevant formulae. ∫_0^1 y (dx/dt) dt = ∫_0^1 (t - t³) (2t) dt = ∫_0^1 (2t² - 2t⁴) dt = [2t³/3 - 2t⁵/5]_0^1 = 2/3 - 2/5 = 4/15
6. [M1A1] ∫_0^1 xy (dx/dt) dt = ∫_0^1 t² (t - t³) (2t) dt = ∫_0^1 (2t⁴ - 2t⁶) dt = [2t⁵/5 - 2t⁷/7]_0^1 = 2/5 - 2/7 = 4/35
7. [M1A1] 1/2 ∫_0^1 y² (dx/dt) dt = 1/2 ∫_0^1 (t - t³)² (2t) dt = ∫_0^1 (t² - 2t⁴ + t⁶) t dt = [t⁴/4 - 2t⁶/6 + t⁸/8]_0^1 = 1/4 - 1/3 + 1/8 = 1/24
8. [A1] Centroid is (4/7, 55/192) Or (0.571,0.286)