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GATE 2014 CS (CS3) – Question 56

Engineering Mathematics · Calculus · 2 marks · Multiple choice

For numerical evaluation of $K=\int_a^b x^2\,dx$, where a and b are given, assess: I. The trapezoidal-rule value is always at least the exact integral. II. The Simpson’s-rule value always equals the exact integral.

  1. I only
  2. II only
  3. Both I and II
  4. Neither I nor II

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Correct answer: (C) Both I and II

Explanation

For a<b, x²is convex, so trapezoidal chords lie above the curve and overestimate its integral. Simpson’s rule integrates all polynomials through degree3exactly, hence alsox². Thus **bothIandII,C**. The inequality assumes the conventional increasing integration bounds.