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4PM1/02R Jun 2023, Question 7: Volumes of Revolution

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2R · Calculator · 7 marks

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The question, in short

Given the curve y = (cos 3θ + √3 sin 3θ)^(1/2) between its x-axis crossings at m and n, find the exact intercepts and the volume when the shaded region is rotated 2π about the θ-axis.

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

Part (a) · 3 marks

Determine the exact coordinates m and n where the curve meets the horizontal axis.

  1. 1cos 3θ + √3 sin 3θ = 0
  2. 2tan 3θ = −1/√3
  3. 33θ = −π/6 or 5π/6
  4. 4m = −π/18, n = 5π/18

Answerm = −π/18, n = 5π/18

Part (b) · 4 marks

Using calculus, calculate the exact volume of the solid formed by rotating region R fully about the θ-axis.

  1. 1V = π∫ from −π/18 to 5π/18 of (cos 3θ + √3 sin 3θ) dθ
  2. 2= π[sin 3θ/3 − √3 cos 3θ/3] from −π/18 to 5π/18
  3. 3Upper: (1/2)/3 − √3(−√3/2)/3 = 1/6 + 1/2 = 2/3
  4. 4Lower: (−1/2)/3 − √3(√3/2)/3 = −1/6 − 1/2 = −2/3
  5. 5V = π(2/3 + 2/3) = 4π/3

Answer4π/3

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