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WMA12/01 Oct 2021, Question 3: The Trapezium Rule

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

The question, in short

Use the trapezium rule with four equal strips on y = log₁₀ x from x = 2 to x = 14 to approximate the shaded area, then adapt that result for two related integrals.

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

Part (a) · 3 marks

Show the trapezium rule with four strips gives approximately 10.10 for the area under log₁₀ x between x = 2 and x = 14.

  1. 1h = (14 − 2)/4 = 3, so x = 2, 5, 8, 11, 14
  2. 2y-values: 0.3010, 0.6990, 0.9031, 1.0414, 1.1461
  3. 3Area ≈ (3/2)[0.3010 + 1.1461 + 2(0.6990 + 0.9031 + 1.0414)]
  4. 4= 1.5 × 6.7342 = 10.10

AnswerShown: (3/2)[log 2 + log 14 + 2(log 5 + log 8 + log 11)] = 10.10

Part (b) · 1 mark

Explain how to improve the accuracy of the trapezium rule estimate for this area.

  1. 1Narrower strips make each trapezium fit the curve more closely
  2. 2So increase the number of strips

AnswerUse more strips (smaller strip width)

Part (ci) · 1 mark

Use the part (a) value to estimate the integral of log₁₀ √x from 2 to 14.

  1. 1log₁₀ √x = ½ log₁₀ x
  2. 2∫₂¹⁴ log₁₀ √x dx ≈ ½ × 10.10 = 5.05

Answer5.05

Part (cii) · 3 marks

Use the part (a) value to estimate the integral of log₁₀ 100x³ from 2 to 14, showing the method.

  1. 1log₁₀ 100x³ = log₁₀ 100 + 3 log₁₀ x = 2 + 3 log₁₀ x
  2. 2∫₂¹⁴ 2 dx = 2 × 14 − 2 × 2 = 24
  3. 33 × 10.10 = 30.30
  4. 4Total ≈ 24 + 30.30 = 54.30

Answer54.30 (or 54.3)

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Also searched as: WMA12 October 2021 · WMA12/01 · wma12 oct21 · IAS Maths Pure Mathematics 2 October 2021 · question 3.
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