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8 marksIntegration I

CAPE Pure Mathematics Unit 1 · May/June 2000 · Paper 2 · Question 5(b)(i)

The length, L, of a rod decreases at the rate of 1/(1 + t²) metres per second. It was 10 metres long at time t = 0.

Write down an appropriate differential equation connecting the length of the rod and the rate of decrease at time, t. Hence, derive an expression for L.

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Other parts of this question

  1. 5(a)With the aid of a diagram, show that for n = 4, ∫₀ᵇ f(x)dx ≈ (d/2)[y₀ + 2y₁ + 2y₂ + 2y₃ + y₄], where y₀ = f(a), y₁ = f(a + d), ..., y₄ = f(a + 4d).[6 marks]
  2. 5(b)(ii)Evaluate the length, L, of the rod using the trapezium rule for five ordinates and strips of length 1 unit.[7 marks]
  3. 5(b)(iii)Find the time at which the rate of decrease vanishes.[2 marks]
  4. 5(b)(iv)Show that the rate of decrease is never negative.[2 marks]

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