A-LevelFurther MathematicsIntegration9231/12 · Oct/Nov 2016 · Q711 Marks
The curve C has equation y = e⁻²ˣ. Find, giving your answers correct to 3 significant figures, (i) the mean value of dy/dx over the interval 0 ≤ x ≤ 2, (ii) the coordinates of the centroid of the region bounded by C, x = 0, x = 2 and y = 0.
- (i)the mean value of dy/dx over the interval 0 ≤ x ≤ 2,[2]
- (ii)the coordinates of the centroid of the region bounded by C, x = 0, x = 2 and y = 0.[9]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2016). The mark or mark type is shown in brackets.
- [M1A1] (i) dy/dx = -2e⁻²ˣ. M.V. = ∫₀² (-2e⁻²ˣ)dx / (2-0) = [e⁻²ˣ]₀² / 2 = (e⁻⁴ - 1) / 2 (= -0.491)
- [M1M1] (ii) Coordinates are x̄ = (∫₀² xe⁻²ˣ dx) / (∫₀² e⁻²ˣ dx) and ȳ = (1/2 ∫₀² (e⁻²ˣ)² dx) / (∫₀² e⁻²ˣ dx) (or ȳ = 1/2∫₀² e⁻⁴ˣ dx / ∫₀² e⁻²ˣ dx)
- [B1] ∫₀² e⁻²ˣ dx = [-1/2 e⁻²ˣ]₀² = 1/2(1-e⁻⁴) (= 0.4908...)
- [M1A1] ∫₀² xe⁻²ˣ dx = [-1/2 xe⁻²ˣ]₀² - ∫₀² (-1/2 e⁻²ˣ) dx = -1/2 (2e⁻⁴) + [-1/4 e⁻²ˣ]₀² = -e⁻⁴ -1/4 e⁻⁴ + 1/4 = 1/4 - 5/4 e⁻⁴ (= 0.2271..)
- [M1A1] ∫₀² e⁻⁴ˣ dx = [-1/4 e⁻⁴ˣ]₀² = 1/4(1-e⁻⁸) (= 0.12495...)
- [A1A1] x̄ = 0.463, ȳ = 0.255 (Final answers in algebraic form SRA1)
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