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4PM1/02 Jun 2024, Question 6: Volumes of Revolution

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 10 marks

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The question, in short

Two curves y = 1/(4x) and y = 2x² meet at A; region R is bounded by both curves and y = 4. Find A, then the exact volume when R rotates 360° about the y-axis.

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

Part (a) · 3 marks

Find the coordinates of the intersection point A of the two given curves.

  1. 12x² = 1/(4x)
  2. 2x³ = 1/8, so x = 1/2
  3. 3y = 2 × (1/2)² = 1/2

AnswerA(1/2, 1/2)

Part (b) · 7 marks

Using algebraic integration, find the exact volume of the solid generated when R is rotated 360° about the y-axis.

  1. 1For S: x² = y/2; for C: x² = 1/(16y²)
  2. 2V = π ∫ from 1/2 to 4 of (y/2) dy − π ∫ from 1/2 to 4 of 1/(16y²) dy
  3. 3= π[y²/4] from 1/2 to 4 − π[−1/(16y)] from 1/2 to 4
  4. 4= (π/4)(16 − 1/4) + (π/16)(1/4 − 2)
  5. 5= 63π/16 − 7π/64 = 245π/64

Answer245π/64

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