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WMA12/01 Oct 2022, Question 6: Definite Integrals and Areas

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 7 marks

The question, in short

Approximate the integral of f(x) from 0 to 2 using the trapezium rule with the given table, then estimate the shaded percentage of a 2 by 3 logo.

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Part (a) · 3 marks

Apply the trapezium rule with all tabulated y values from x = 0 to 2 to approximate the integral of f(x), giving the result to 3 decimal places.

  1. 1h = 0.5
  2. 2Area ≈ (0.5/2) × [3 + 1.92 + 2(2.6833 + 2.4 + 2.1466)]
  3. 3= 0.25 × [4.92 + 14.4598] = 0.25 × 19.3798
  4. 4= 4.845 (3 d.p.)

Answer4.845

Part (b) · 4 marks

Using calculus and the part (a) value, find the percentage of the 2 by 3 rectangle occupied by region R between C₁ and y = 2 − (1/4)x².

  1. 1Area under C₂: ∫ from 0 to 2 of (2 − x²/4) dx = [2x − x³/12] from 0 to 2
  2. 2= 4 − 8/12 = 10/3
  3. 3Area of R ≈ 4.845 − 10/3 = 1.5117
  4. 4Rectangle area = 2 × 3 = 6
  5. 5Percentage shaded ≈ 1.5117/6 × 100 = 25.2%

Answer25.2%

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