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GATE 2018 ME (ME1) – Question 13

Engineering Mathematics · Calculus: Definite and improper integrals, double and triple integrals · 1 mark · Multiple choice

For continuous f on [a,b], the integral mean-value theorem gives some ξ in the interval such that $\int_a^bf(x)\,dx$ equals ______.

  1. $f(\xi)(b-a)$
  2. $f(b)(\xi-a)$
  3. $f(a)(b-\xi)$
  4. 0

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Show answer and explanation

Correct answer: (A) $f(\xi)(b-a)$

Explanation

Continuity ensures the average value $\frac1{b-a}\int_a^bf(x)\,dx$ is attained at some ξ. Rearranging yields $\int_a^bf(x)\,dx=f(\xi)(b-a)$ (A).