4PM1/02R Jun 2019, Question 9: Volumes of Revolution
Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2R · Calculator · 9 marks
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The question, in short
Region R is bounded by the y-axis, the line y + 2x = 13 and the curve y = x² − 2 for x ≥ 0. Rotating R through 360° about the y-axis, find the solid's volume in terms of π using algebraic integration.
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Solution · 9 marks
- 1x² − 2 = 13 − 2x ⇒ x² + 2x − 15 = 0 ⇒ (x + 5)(x − 3) = 0
- 2x = 3, y = 7
- 3Cone part (y from 7 to 13): (1/3) × π × 3² × (13 − 7) = 18π
- 4Curve part: π ∫ from −2 to 7 of (y + 2) dy = π[y²/2 + 2y] from −2 to 7
- 5= π{(49/2 + 14) − (2 − 4)} = 81π/2
- 6Total volume = 18π + 81π/2 = 117π/2
Answer117π/2
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