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6 marksIntegration I

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

The area under the curve y = f(x) represented by ∫₀ᵇ f(x)dx is divided into n strips, each of width d units, such that each strip is approximately a trapezium.

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).

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Other parts of this question

  1. 5(b)(i)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.[8 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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