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4PM1/02 Jun 2022, 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

Derive a double-angle identity for cos 2A from a given formula, then use calculus to find the volume when region R is rotated 360° about the x-axis, giving the nearest integer.

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

Part (a) · 2 marks

Starting from a formula on page 2, prove that cos 2A equals 1 minus 2sin²A.

  1. 1cos 2A = cos(A + A) = cos²A − sin²A
  2. 2= (1 − sin²A) − sin²A = 1 − 2sin²A

Answercos 2A = 1 − 2sin²A shown

Part (b) · 6 marks

Region R is bounded by y = 3 + 2sin x, the axes and x = π/4; find the volume of revolution about the x-axis to the nearest integer.

  1. 1V = π∫ from 0 to π/4 of (3 + 2sin x)² dx
  2. 2= π∫ (9 + 12sin x + 4sin²x) dx
  3. 3Using sin²x = (1 − cos 2x)/2: = π∫ (11 + 12sin x − 2cos 2x) dx
  4. 4= π[11x − 12cos x − sin 2x] from 0 to π/4
  5. 5= π[(11π/4 − 6√2 − 1) − (−12)] = 35.04…
  6. 6V ≈ 35

Answer35

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