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4PM1/01R Jun 2022, Question 10: Definite Integrals and Areas

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1R · Calculator · 11 marks

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The question, in short

Use the given trigonometric formulae to prove an identity, then use calculus on the curve y = 6sin 4x cos x to find the total area between x = 0 and x = π/2, to 3 significant figures.

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Part (a) · 3 marks

Show that 2sin 4x cos x equals sin 5x + sin 3x, using the sin(A + B) and sin(A − B) results.

  1. 1sin(A + B) + sin(A − B) = 2sin A cos B
  2. 2Let A = 4x, B = x
  3. 3sin 5x + sin 3x = 2sin 4x cos x

Answer2sin 4x cos x = sin 5x + sin 3x shown

Part (b) · 8 marks

Using calculus, find the total area enclosed by y = 6sin 4x cos x and the x-axis from x = 0 to x = π/2, to 3 significant figures.

  1. 1y = 0 ⇒ sin 4x = 0 or cos x = 0: x = 0, π/4, π/2 in the interval
  2. 2y = 3(sin 5x + sin 3x); ∫ = 3(−(cos 5x)/5 − (cos 3x)/3)
  3. 3From 0 to π/4: area = (8 + 4√2)/5 = 2.731…
  4. 4From π/4 to π/2: integral = −4√2/5, area = 1.131… (curve below the axis)
  5. 5Total area = 2.731 + 1.131 = 3.86

Answer3.86

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