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WMA12/01 Oct 2019, Question 5: The Trapezium Rule

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

The question, in short

Sketch y = aˣ for 0 < a < 1 with the y-axis intercept, then use the given table with the trapezium rule to estimate two related definite integrals from 2 to 4.

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

Part (a) · 2 marks

Sketch the curve y = aˣ where 0 < a < 1, marking the coordinates where it crosses the y-axis.

  1. 1For 0 < a < 1, aˣ decreases as x increases and is always positive
  2. 2At x = 0, y = a⁰ = 1, so the curve crosses the y-axis at (0, 1)
  3. 3As x → ∞, y → 0 (the x-axis is an asymptote); as x → −∞, y grows without limit

AnswerDecreasing exponential curve above the x-axis in quadrants 1 and 2 only, crossing the y-axis at (0, 1); it gets steeper as x → −∞ and approaches the x-axis (y = 0) as x → ∞ without touching it

Part (b) · 3 marks

Apply the trapezium rule to all tabulated y values to estimate the integral of x² + (1/2)ˣ from 2 to 4.

  1. 1Strip width h = 0.5
  2. 2Area ≈ (0.5/2){4.25 + 16.06 + 2(6.427 + 9.125 + 12.34)}
  3. 3= 0.25 × {20.31 + 55.784} = 0.25 × 76.094
  4. 4≈ 19.0235 ≈ 19.0

Answer19.0 (19.0235)

Part (c) · 2 marks

Using the part (b) result, estimate the integral of x(x − 3) + (1/2)ˣ from 2 to 4, showing your method.

  1. 1x(x − 3) + (1/2)ˣ = x² + (1/2)ˣ − 3x
  2. 2So the integral = (answer to b) − ∫₂⁴ 3x dx
  3. 3∫₂⁴ 3x dx = [3x²/2]₂⁴ = 24 − 6 = 18
  4. 4Estimate ≈ 19.0235 − 18 = 1.0235 ≈ 1.0

Answer1.0 (1.0235)

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Also searched as: WMA12 October 2019 · WMA12/01 · wma12 oct19 · IAS Maths Pure Mathematics 2 October 2019 · question 5.
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