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4PM1/01R Jan 2022, Question 11: Volumes of Revolution

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

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The question, in short

Curve y = e^(2x) − 9 meets the axes at (0, a) and (b, 0), enclosing region R. Determine a, verify b = ln 3, then find the volume when R rotates fully about the x-axis, giving π(p ln 3 + q).

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

Part (a) · 3 marks

For the intercepts, work out the y-value a where x = 0, and demonstrate that the x-intercept equals ln 3.

  1. 1x = 0: y = e⁰ − 9 = −8, so a = −8
  2. 2y = 0: e^(2x) = 9
  3. 32x = ln 9
  4. 4x = ½ ln 9 = ln 3

Answer(i) a = −8 (ii) b = ln 3 shown

Part (b) · 6 marks

Using calculus, calculate the solid's volume from rotating R 360° about the x-axis, expressed as π(p ln 3 + q) with integers p and q.

  1. 1V = π ∫ from 0 to ln 3 of (e^(2x) − 9)² dx
  2. 2= π ∫ from 0 to ln 3 of (e^(4x) − 18e^(2x) + 81) dx
  3. 3= π [¼e^(4x) − 9e^(2x) + 81x] from 0 to ln 3
  4. 4= π[(81/4 − 81 + 81 ln 3) − (1/4 − 9)]
  5. 5= π(81 ln 3 − 52)

Answerπ(81 ln 3 − 52)

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