A-LevelFurther MathematicsIntegration9231/12 · May/June 2011 · Q912 Marks

The curve C has equation \( y = x^{3/2} \).

Find the coordinates of the centroid of the region bounded by C, the lines \( x = 1 \), \( x = 4 \) and the x-axis.[7]
Show that the length of the arc of C from the point where \( x = 5 \) to the point where \( x = 28 \) is 139.[5]

Mark scheme

Marking points from the official mark scheme (May/June 2011). The mark or mark type is shown in brackets.

[M1M1] Uses formulae for coords of centroid. barx = (∫ x y dx)/(∫ y dx) , bary = (1/2 ∫ y² dx)/(∫ y dx)
[A1A1] Numerators. ∫₁⁴ x^(5/2) dx = [ 2/7 x^(7/2) ]₁⁴ = 2/7 (128 - 1) = 254/7 1/2 ∫₁⁴ x³ dx = 1/2 [ 1/4 x⁴ ]₁⁴ = 1/8 (256 - 1) = 255/8
[B1] Denominator. ∫₁⁴ x^(3/2) dx = [ 2/5 x^(5/2) ]₁⁴ = 2/5 (32 - 1) = 62/5
[M1A1] Obtains values. (Accept rational values.) barx = (254/7)/(62/5) = 1270/434 = 635/217 ≈ 2.93 bary = (255/8)/(62/5) = 1275/496 ≈ 2.57
[B1] Differentiates y wrt x and squares. y = x^(3/2) ⇒ y' = 3/2 x^(1/2) = 3/2 √x ⇒ (y')² = 9/4 x
[B1] Subs in arc length formula (LR). S = ∫₅²⁸ √(1 + 9/4 x) dx
[M1A1] Integrates. = 8/27 [ (1 + 9/4 x)^(3/2) ]₅²⁸ or 1/27 [ (4 + 9x)^(3/2) ]₅²⁸
[A1] Obtains printed result. = 8/27 (84^(3/2) - 7^(3/2)) or 1/27 (372^(3/2) - 49^(3/2)) = 139 (AG)

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