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4PM1/01 Jan 2020, Question 5: Definite Integrals and Areas

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

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The question, in short

Given curve y = 25 − x² and line y + x = 13 meeting at A and B, find the intersection coordinates and the shaded area R bounded by curve, line and x-axis.

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

Part (a) · 4 marks

Solve the curve and line equations simultaneously to find the coordinates of points A and B.

  1. 125 − x² = 13 − x
  2. 2x² − x − 12 = 0
  3. 3(x − 4)(x + 3) = 0, so x = −3 or x = 4
  4. 4x = −3: y = 16; x = 4: y = 9
  5. 5A = (−3, 16), B = (4, 9)

AnswerA = (−3, 16), B = (4, 9)

Part (b) · 7 marks

Use algebraic integration to calculate the area of region R bounded by the curve, line and x-axis.

  1. 1R = ∫ from −5 to −3 of (25 − x²) dx + trapezium under AB + ∫ from 4 to 5 of (25 − x²) dx
  2. 2Trapezium from x = −3 to x = 4: ½(16 + 9) × 7 = 87.5
  3. 3∫ (25 − x²) dx = 25x − x³/3
  4. 4∫ from −5 to −3 = (−75 + 9) − (−125 + 125/3) = 52/3
  5. 5∫ from 4 to 5 = (125 − 125/3) − (100 − 64/3) = 14/3
  6. 6Area = 52/3 + 87.5 + 14/3 = 109.5

Answer219/2 (109.5)

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