◈ Oracle PrepSign Up Free
A-LevelFurther MathematicsIntegration9231/12 · Oct/Nov 2012 · Q26 Marks

The curve C has equation y = 2x² for 0 ≤ x ≤ 4. Find

  1. (i)the mean value of y with respect to x for 0 ≤ x ≤ 4,[3]
  2. (ii)the y-coordinate of the centroid of the region enclosed by C, the line x = 4 and the x-axis.[3]

Mark scheme

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

  1. [M1] Uses formula for mean value: (1/4) ∫ (from 0 to 4) 2x² dx
  2. [M1A1] Integrates: (1/4) [2x³/3] (from 0 to 4) = 32/3
  3. [M1] Uses formula for y-coordinate: (1/2) ∫ (from 0 to 4) y² dx / ∫ (from 0 to 4) y dx
  4. [M1A1] Integrates: (1/2) [4x⁵/5] (from 0 to 4) / [2x³/3] (from 0 to 4) = 48/5

Write your answer and get it marked

On Oracle Prep you can write or photograph your own answer and have it marked by AI against this mark scheme, with feedback on what you missed.

Sign up free →

About this question

Topic
Integration, A-Level Further Mathematics (syllabus 9231). Structured question, 6 marks.
Source
A-Level Further Mathematics, Oct/Nov 2012 Paper 1 Variant 2, question 2.

Questions and mark schemes are from past examination papers and remain the copyright of the exam board. How Oracle Prep builds its content

Related Further Mathematics questions

IntegrationMay/June 20177m

Let I_n = ∫ (from 0 to ½π) xⁿ sin x dx.

IntegrationMay/June 201714m

Answer only one of the following two alternatives. EITHER The curve C has equation y = ½(eˣ + e⁻ˣ) for 0 ≤ x ≤ 4.

IntegrationMay/June 20185m

The curve C is defined parametrically by x = eᵗ - t, y = 4e½ᵗ. Find the length of the arc of C from the point where t…

Roots of polynomial equationsMay/June 20164m

The roots of the cubic equation 2x³ + x² − 7 = 0 are α, β and γ. Using the substitution y = 1 + (1/x), or otherwise, …

Summation of seriesMay/June 20166m

Express 4/(r(r + 1)(r + 2)) in partial fractions and hence find sum from r=1 to n of 4/(r(r + 1)(r + 2)). Deduce the …

Proof by inductionMay/June 20166m

Prove by mathematical induction that, for all positive integers n, 10ⁿ + 3 × 4ⁿ⁺² + 5 is divisible by 9.

← Browse all past-paper questions

© 2026 Oracle Prep