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TMUA 2021 · Paper 1 · Question 15 of 20

TMUA 2021 Paper 1 Question 15

Differentiation and integration — Periodic graphs · integrating a sum of stretched copies. Try it first; the answer and a full worked solution are below.

TMUA 2021 · Paper 1Differentiation and integrationPeriodic graphs · integrating a sum of stretched copies6 optionshard

The original question includes a diagram: A triangular wave y = f(x) made of straight segments of gradient 1 and -1, taking the value 0 at every even integer and 1 at every odd integer, repeating for all x.

The diagram shows the graph of y = f(x). It consists of alternating straight-line segments of gradient 1 and −1: the graph is 0 at x = 0,  2,  4,  … and rises to 1 at x = 1,  3,  5,  …, continuing in this way for all values of x.

The function g is defined as g(x) = ∑r=110 f(2r−1x)

Find the value of ∫01 g(x) dx

  1. A10231024
  2. B1023512
  3. C5
  4. D10
  5. E552
  6. F55
Show the answer and worked solution
answer · C
  1. A10231024
  2. B1023512
  3. C5
  4. D10
  5. E552
  6. F55
Integration is linear, so work out ∫01 f(kx)dx for each k = 2r−1 separately. Substituting u = kx gives 1k∫0k f(u) du. The wave has period 2 and averages 12 over each period, so for k even ∫0k f = k2 and the term is 12. For r = 1, k = 1 and ∫01 f(u) du is the triangle of base 1 and height 1 cut in half, again 12. Every one of the ten terms is 12, so the total is 5.