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4PM1/02R Jan 2022, Question 10: Definite Integrals and Areas

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2R · Calculator · 12 marks

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The question, in short

For curves y = sin x + 1 and y = cos x + 1 meeting at A and B, find exact x coordinates of the intersections, then use calculus to find the ratio of areas R₁ and R₂.

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

Part (a) · 3 marks

Find the exact x coordinates of the two intersection points A and B of the given curves.

  1. 1sin x + 1 = cos x + 1
  2. 2tan x = 1
  3. 3x = π/4 (A), x = 5π/4 (B)

AnswerA: x = π/4, B: x = 5π/4

Part (b) · 9 marks

Using calculus, find the ratio of the area of R₁ to the area of R₂, giving it in the form a : (π√2/b − c).

  1. 1R₁ = ∫ from π/4 to 5π/4 of (sin x − cos x) dx
  2. 2= [−cos x − sin x] from π/4 to 5π/4 = √2 − (−√2) = 2√2
  3. 3R₂ = ∫ from π to 5π/4 of (cos x + 1) dx + ∫ from 5π/4 to 3π/2 of (sin x + 1) dx
  4. 4= [sin x + x] from π to 5π/4 + [−cos x + x] from 5π/4 to 3π/2
  5. 5= (−√2/2 + π/4) + (π/4 − √2/2) = π/2 − √2
  6. 6R₁ : R₂ = 2√2 : (π/2 − √2)
  7. 7Divide by √2: 2 : (π√2/4 − 1)

Answer2 : (π√2/4 − 1)

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