TTMUA Lab
TMUA 2017 · Paper 1 · Question 12 of 20

TMUA 2017 Paper 1 Question 12

Differentiation and integration — Definite integrals · translation of the variable. Try it first; the answer and a full worked solution are below.

TMUA 2017 · Paper 1Differentiation and integrationDefinite integrals · translation of the variable6 options
The polynomial function f(x) is such that f(x) > 0 for all values of x.

Given ∫24 f(x) dx = A, which one of the following statements must be correct?

  1. A∫02 [f(x+2) + 1] dx = A + 1
  2. B∫02 [f(x+2) + 1] dx = A + 2
  3. C∫24 [f(x+2) + 1] dx = A + 1
  4. D∫24 [f(x+2) + 1] dx = A + 2
  5. E∫46 [f(x+2) + 1] dx = A + 1
  6. F∫46 [f(x+2) + 1] dx = A + 2
Show the answer and worked solution
answer · B
  1. A∫02 [f(x+2) + 1] dx = A + 1
  2. B∫02 [f(x+2) + 1] dx = A + 2
  3. C∫24 [f(x+2) + 1] dx = A + 1
  4. D∫24 [f(x+2) + 1] dx = A + 2
  5. E∫46 [f(x+2) + 1] dx = A + 1
  6. F∫46 [f(x+2) + 1] dx = A + 2
Replacing x by x+2 shifts the graph two units to the left, so an integral of f(x+2) over [a, b] equals the integral of f over [a+2,  b+2]. To recover ∫24 f the limits must therefore be 0 and 2. The added constant contributes 1 × (2−0) = 2, so ∫02 [f(x+2)+1] dx = A + 2. The condition f(x) > 0 is there only to rule out cancellation arguments; it does not change the substitution.