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WMA12/01 Oct 2019, Question 8: Definite Integrals and Areas

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 9 marks

The question, in short

For the curve y = (8√x − 5)/(2x²), x > 0, find the exact area of the shaded region R bounded by the curve, x = 2, x = 4 and the x-axis, without using a calculator.

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

Part (i) · 5 marks

Calculate the exact area of R, the region under the curve between x = 2 and x = 4 and above the x-axis.

  1. 1(8√x − 5)/(2x²) = 4x^(−3/2) − (5/2)x⁻²
  2. 2∫ (4x^(−3/2) − (5/2)x⁻²) dx = −8x^(−1/2) + (5/2)x⁻¹
  3. 3At x = 4: −8/2 + 5/8 = −4 + 5/8
  4. 4At x = 2: −8/√2 + 5/4 = −4√2 + 5/4
  5. 5Area = (−4 + 5/8) − (−4√2 + 5/4) = 4√2 − 37/8

Answer4√2 − 37/8

Part (ii) · 4 marks

Determine the constant k so that the definite integral from −3 to 6 of ((1/2)x² + k) equals 55.

  1. 1∫ ((1/2)x² + k) dx = (1/6)x³ + kx
  2. 2[(1/6)x³ + kx] from −3 to 6 = (36 + 6k) − (−9/2 − 3k)
  3. 3= 81/2 + 9k = 55
  4. 49k = 29/2, so k = 29/18

Answerk = 29/18

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