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WMA12/01 Oct 2020, Question 6: Definite Integrals and Areas

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 9 marks

The question, in short

Given curves C₁: y = x³ − 6x + 9 and C₂: y = −2x² + 7x − 1 for x ≥ 0, with intersection A at (1, 4), find point B and the exact area of the shaded region R between the curves.

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Part (a) · 4 marks

Solve the two curve equations simultaneously by algebra, showing all steps, to find the coordinates of the second intersection point B.

  1. 1x³ − 6x + 9 = −2x² + 7x − 1
  2. 2x³ + 2x² − 13x + 10 = 0
  3. 3x = 1 is a root, so (x − 1)(x² + 3x − 10) = 0
  4. 4(x − 1)(x − 2)(x + 5) = 0, so x = 2 (x = −5 is outside x ≥ 0)
  5. 5y = −2(4) + 14 − 1 = 5, so B is (2, 5)

AnswerB(2, 5)

Part (b) · 5 marks

Set up and evaluate an algebraic definite integral between A and B to find the exact area of the shaded region R.

  1. 1Area = ∫ from 1 to 2 of [(−2x² + 7x − 1) − (x³ − 6x + 9)] dx
  2. 2= ∫ from 1 to 2 of (−x³ − 2x² + 13x − 10) dx
  3. 3= [−x⁴/4 − 2x³/3 + 13x²/2 − 10x] from 1 to 2
  4. 4At x = 2: −4 − 16/3 + 26 − 20 = −10/3; at x = 1: −1/4 − 2/3 + 13/2 − 10 = −53/12
  5. 5Area = −10/3 − (−53/12) = 13/12

Answer13/12

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