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TMUA 2019 · Paper 1 · Question 8 of 20

TMUA 2019 Paper 1 Question 8

Differentiation and integration — Trapezium rule · concavity decides over- or underestimate. Try it first; the answer and a full worked solution are below.

TMUA 2019 · Paper 1Differentiation and integrationTrapezium rule · concavity decides over- or underestimate5 options
The function f is such that 0 < f(x) < 1 for 0 ≤ x ≤ 1.

The trapezium rule with n equal intervals is used to estimate ∫01 f(x) dx and produces an underestimate.

Using the same number of equal intervals, for which one of the following does the trapezium rule produce an overestimate?

  1. A∫01 (f(x) + 1)dx
  2. B∫01 2f(x) dx
  3. C∫−10 f(x+1) dx
  4. D∫−10 f(−x) dx
  5. E∫01 (1 − f(x))dx
Show the answer and worked solution
answer · E
  1. A∫01 (f(x) + 1)dx
  2. B∫01 2f(x) dx
  3. C∫−10 f(x+1) dx
  4. D∫−10 f(−x) dx
  5. E∫01 (1 − f(x))dx
The trapezium rule replaces the curve by chords, so an underestimate means the chords lie below the curve: f curves downwards. To get an overestimate you need an integrand that curves upwards. Adding 1, doubling, translating (option C is ∫12 f(u) du after u = x+1) and reflecting in the y-axis all leave the direction of curvature alone, so those four stay underestimates. Only 1 − f(x) turns the graph upside down, so its chords lie above it and the rule overestimates.