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TMUA 2020 · Paper 2 · Question 6 of 20

TMUA 2020 Paper 2 Question 6

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

TMUA 2020 · Paper 2Differentiation and integrationDefinite integrals · necessary conditions8 options
The function f(x) is defined for all real values of x.

Which of the following conditions on f(x) is/are necessary to ensure that ∫−50f(x) dx = ∫05 f(x) dx

Condition I:    f(x) = f(−x) for −5 ≤ x ≤ 5
Condition II:   f(x) = c for −5 ≤ x ≤ 5, where c is a constant
Condition III: f(x) = −f(−x) for −5 ≤ x ≤ 5

  1. Anone of them
  2. BI only
  3. CII only
  4. DIII only
  5. EI and II only
  6. FI and III only
  7. GII and III only
  8. HI, II and III
Show the answer and worked solution
answer · A
  1. Anone of them
  2. BI only
  3. CII only
  4. DIII only
  5. EI and II only
  6. FI and III only
  7. GII and III only
  8. HI, II and III
Read the word necessary carefully: a condition is necessary only if it holds every time the two integrals are equal, so a single function with equal integrals that breaks the condition rules it out. One such function does for all three at once. Take f(x) = x3 − 252x + 1. Its odd part x3 − 252x integrates to 6254 − 6254 = 0 over 0 ≤ x ≤ 5, so both of the required integrals are just the contribution of the constant, namely 5, and they are equal. Yet this f is not even, not constant and not odd. So none of the conditions is necessary. Conditions I and II are sufficient, which is the answer this question is set up to tempt you into.