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4PM1/02 Jan 2020, Question 6: Volumes of Revolution

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 8 marks

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The question, in short

Given a curve and a line meeting at O and A, prove A's coordinates, then find the volume when the enclosed region R is rotated fully about the x-axis, giving the result in terms of π.

Summarised in our own words. Use the official paper for the exact wording and the diagram.

Part (a) · 2 marks

By solving the curve and line equations together, demonstrate that their non-origin intersection A is (3/2, 3/4).

  1. 1Line: y = x/2. Set 2x − x² = x/2
  2. 2x² − (3/2)x = 0, so x = 0 or x = 3/2
  3. 3x = 3/2 gives y = 3/4
  4. 4A = (3/2, 3/4)

Answer(3/2, 3/4) lies on both the curve and the line

Part (b) · 6 marks

Using algebraic integration of the difference of squared functions between the x-limits, calculate the solid's volume in terms of π.

  1. 1Volume under curve = π ∫ from 0 to 3/2 of (2x − x²)² dx = π ∫ (4x² − 4x³ + x⁴) dx
  2. 2= π[4x³/3 − x⁴ + x⁵/5] from 0 to 3/2 = 153π/160
  3. 3Cone from the line: π ∫ from 0 to 3/2 of (x/2)² dx = 9π/32
  4. 4Volume = 153π/160 − 45π/160 = 27π/40

Answer27π/40

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