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4PM1/02R Jan 2020, Question 10: Integration: Undoing Differentiation

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2R · Calculator · 12 marks

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The question, in short

Given a curve's gradient 3x² − 4x − p passing through (2, 0) and (−1, 9), find p and the curve's equation, then the area between the curve and line y = 8 − 4x.

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Part (a) · 6 marks

Determine p using the two given points, then integrate to obtain the curve's equation in the stated form.

  1. 1y = ∫ (3x² − 4x − p) dx = x³ − 2x² − px + c
  2. 2(2, 0): 0 = 8 − 8 − 2p + c, so c = 2p
  3. 3(−1, 9): 9 = −1 − 2 + p + c, so c = 12 − p
  4. 42p = 12 − p gives p = 4, c = 8
  5. 5y = x³ − 2x² − 4x + 8

Answery = x³ − 2x² − 4x + 8 shown (p = 4, c = 8)

Part (b) · 6 marks

Integrate algebraically to calculate the exact area of the finite region enclosed by the curve and the line.

  1. 1x³ − 2x² − 4x + 8 = 8 − 4x gives x³ − 2x² = 0
  2. 2x²(x − 2) = 0, so x = 0 or x = 2
  3. 3Area = ∫ from 0 to 2 of [(8 − 4x) − (x³ − 2x² − 4x + 8)] dx = ∫ from 0 to 2 of (2x² − x³) dx
  4. 4= [2x³/3 − x⁴/4] from 0 to 2 = 16/3 − 4
  5. 5= 4/3

Answer4/3

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