The curve C has equation y = 2(x/3)^(3/2), where 0 ≤ x ≤ 3. Show that the arc length of C is 2(2√2 – 1). Find the coordinates of the centroid of the region enclosed by C, the x-axis and the line x = 3.

Mark scheme

Marking points from the official mark scheme (May/June 2012). The mark or mark type is shown in brackets.

1. [B1] Differentiates and squares. y' = (x/3)^(1/2) ⇒ (y')² = x/3
2. [M1] Uses formula for arc length. s = ∫√(1 + x/3) dx
3. [A1A1] Integrates and obtains value. = [2(1+x/3)^(3/2) / (3/2) * (1/3)] evaluated from 0 to 3 = 2(2√2 – 1) (AG)
4. [M1] Uses formula for x-coordinate of centroid. x̄ = (∫x y dx) / (∫y dx)
5. [A1A1] Integrates both expressions and obtains value. x̄ = 15/7 (= 2.14)
6. [M1] Uses formula for y-coordinate of centroid. ȳ = (½ ∫y² dx) / (∫y dx)
7. [A1A1] Integrates both expressions and obtains value. ȳ = 5/8 (= 0.625)