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WMA12/01 Oct 2023, Question 9: Definite Integrals and Areas

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 12 marks

The question, in short

For the curve y = (2/3)x² − 9√x + 13, x ≥ 0, use calculus to find where y increases, then find the exact area of region R bounded by the curve, the tangent at P(9, 40), and both axes.

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

Part (a) · 4 marks

Differentiate the given curve and solve for the x values, x ≥ 0, where y is increasing.

  1. 1dy/dx = (4/3)x − (9/2)x^(−1/2)
  2. 2Increasing when (4/3)x − 9/(2√x) > 0
  3. 3Multiply by 6√x (positive): 8x^(3/2) > 27, so x^(3/2) > 27/8
  4. 4x > (27/8)^(2/3) = 9/4

Answerx > 9/4

Part (b) · 8 marks

Find the tangent line at P(9, 40), then use calculus to calculate the exact area of the shaded region R.

  1. 1∫ ((2/3)x² − 9x^(1/2) + 13) dx = (2/9)x³ − 6x^(3/2) + 13x
  2. 2Area under C from 0 to 9 = (2/9)(729) − 6(27) + 117 = 162 − 162 + 117 = 117
  3. 3At x = 9: dy/dx = 12 − 9/6 = 21/2, so l: y − 40 = (21/2)(x − 9)
  4. 4l meets the x-axis where −40 = (21/2)(x − 9), so x = 9 − 80/21 = 109/21
  5. 5Triangle under l from x = 109/21 to 9: (1/2) × 40 × 80/21 = 1600/21
  6. 6Area of R = 117 − 1600/21 = 857/21

AnswerArea of R = 857/21

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