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WMA12/01 Jun 2022, Question 2: The Trapezium Rule

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 8 marks

The question, in short

Complete a table of y values for y = 1 − log₁₀(sin x) at given x, then estimate two definite integrals over 0.5 to 3 using the trapezium rule and the first result.

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

Part (a) · 2 marks

Calculate the missing y values at x = 1 and x = 2, giving each to 3 decimal places.

  1. 1Use radians: sin 1 = 0.8415, so y = 1 − log₁₀(0.8415) = 1.075
  2. 2sin 2 = 0.9093, so y = 1 − log₁₀(0.9093) = 1.041

AnswerWhen x = 1, y = 1.075; when x = 2, y = 1.041

Part (b) · 3 marks

Apply the trapezium rule with all six y values and h = 0.5 to estimate the integral of 1 − log₁₀(sin x) from 0.5 to 3, to 2 decimal places.

  1. 1Strip width h = 0.5
  2. 2Area ≈ (0.5/2) × [1.319 + 1.850 + 2(1.075 + 1.001 + 1.041 + 1.223)]
  3. 3= 0.25 × [3.169 + 2 × 4.340] = 0.25 × 11.849
  4. 4= 2.962… ≈ 2.96

Answer2.96

Part (c) · 3 marks

Using the part (b) estimate, find the integral of 3 + log₁₀(sin x) from 0.5 to 3.

  1. 1Write 3 + log₁₀(sin x) = 4 − (1 − log₁₀(sin x))
  2. 2∫ from 0.5 to 3 of 4 dx = 4 × (3 − 0.5) = 10
  3. 3So the integral ≈ 10 − 2.96 = 7.04

Answer7.04

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