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With respect to the numerical evaluation of the definite integral, K = ∫ₐ^b x² dx, where a and b are given, which of the following statements is/are TRUE? I) The value of K obtained using the trapezoidal rule is always greater than or equal to the exact value of the definite integral. II) The value of K obtained using the Simpson’s rule is always equal to the exact value of the definite integral.

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

Correct answer: Both I and II

Solution

For the convex function x^2 the trapezoidal rule overestimates, so K_trap >= exact (I true). Simpson's rule integrates polynomials up to degree 3 exactly, so it gives the exact value for x^2 (II true). Hence both I and II are true.

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