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TMUA 2021 · Paper 1 · Question 13 of 20

TMUA 2021 Paper 1 Question 13

Differentiation and integration — Definite integrals · building a sum from unit intervals. Try it first; the answer and a full worked solution are below.

TMUA 2021 · Paper 1Differentiation and integrationDefinite integrals · building a sum from unit intervals6 options
The function f is such that, for every integer n ∫nn+1 f(x) dx = n + 1 Evaluate ∑r=18(∫0r f(x) dx)
  1. A36
  2. B84
  3. C120
  4. D165
  5. E204
  6. F288
Show the answer and worked solution
answer · C
  1. A36
  2. B84
  3. C120
  4. D165
  5. E204
  6. F288
Chop each integral into unit pieces: ∫0r f = ∑n=0r−1(n+1) = 1 + 2 + ⋯ + r = r(r+1)2. So the required sum is ∑r=18r(r+1)2 = 12(∑ r2 + ∑ r). With ∑r=18 r2 = 204 and ∑r=18 r = 36, this is 12(240) = 120.