4PM1/01 Nov 2023, Question 5: Volumes of Revolution
Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 8 marks
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The question, in short
Find the exact volume, in terms of π, of the solid formed when the region between y = √(2x + 6) and 3y − x = 3 is rotated 360° about the x-axis, using algebraic integration.
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Solution · 8 marks
- 1√(2x + 6) = x/3 + 1, so 2x + 6 = x²/9 + 2x/3 + 1
- 2x² − 12x − 45 = 0, (x + 3)(x − 15) = 0, x = −3, 15
- 3At x = 15, y = 6
- 4Volume under curve: π∫ from −3 to 15 of (2x + 6) dx = π[x² + 6x] from −3 to 15 = 324π
- 5Cone under the line: (1/3)π(6²)(18) = 216π
- 6V = 324π − 216π = 108π
Answer108π
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