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7 marksThe Real Number System – ℝ

CAPE Pure Mathematics Unit 1 · May/June 2003 · Paper 2 · Question 6(a)

In a diagram not drawn to scale, the area under the curve y = (1 + x)⁻¹, 0 ≤ x ≤ 1, is approximated by a set of n rectangular strips each of width 1/n units.

Show that the sum, S, of the areas of the rectangular strips is 1/(n+1) + 1/(n+2) + ... + 1/(2n).

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Other parts of this question

  1. 6(b)(i)Show that for f(x) = x/(x² + 4), f'(x) = (4 - x²)/(x² + 4)².[4 marks]
  2. 6(b)(ii)Hence, evaluate ∫₀² (12 - 3x²)/(x² + 4)² dx.[4 marks]
  3. 6(c)(i)Sketch the curve y = x² + 1.[3 marks]
  4. 6(c)(ii)Find the volume obtained by rotating the portion of the curve between x = 0 and x = 1 through 2π radians about the y axis.[7 marks]

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