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4PM1/01 Jan 2023, Question 10: Volumes of Revolution

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 9 marks

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The question, in short

Figure 2 shows region R bounded by y = x² + 1 and x² + y² = 11, intersecting at A and B. Find the x coordinates of A and B, then the volume when R rotates 360° about the x-axis.

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

Part (a) · 4 marks

Find the x coordinate of A and the x coordinate of B where the parabola and circle meet.

  1. 1x² + (x² + 1)² = 11
  2. 2x⁴ + 3x² − 10 = 0
  3. 3(x² + 5)(x² − 2) = 0, so x² = 2
  4. 4A: x = −√2, B: x = √2

AnswerA: x = −√2, B: x = √2

Part (b) · 5 marks

Using algebraic integration, find the volume of the solid formed when R is rotated 360° about the x-axis, to 2 decimal places.

  1. 1V = π ∫ from −√2 to √2 of [(11 − x²) − (x² + 1)²] dx
  2. 2= π ∫ from −√2 to √2 of (10 − 3x² − x⁴) dx
  3. 3= π [10x − x³ − x⁵/5] from −√2 to √2
  4. 4= 2π(10√2 − 2√2 − 4√2/5) = 72√2π/5
  5. 5= 63.98

Answer63.98

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