GATE CS 2014 Set 3 — Question 56
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Engineering Mathematics → Calculus → Integration
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Question
With respect to the numerical evaluation of the definite integral, , where and are given, which of the following statements is/are TRUE?I) The value of obtained using the trapezoidal rule is always greater than or equal to the exact value of the definite integral.
II) The value of obtained using the Simpson's rule is always equal to the exact value of the definite integral.
II) The value of obtained using the Simpson's rule is always equal to the exact value of the definite integral.
Correct answer
(C) Both I and II
Solution
The function to be integrated is .Statement I: The Trapezoidal rule approximates the area under the curve by trapezoids. For a function with (concave up or convex) on the interval , the trapezoidal rule overestimates the integral. Since , , the graph is concave up. Thus, the trapezoidal approximation will be greater than the exact area. The statement says "greater than or equal to", which is true.Statement II: Simpson's rule (specifically Simpson's 1/3 rule) is exact for polynomials of degree up to 3. Since the integrand is a polynomial of degree 2 (), Simpson's rule will provide the exact value of the integral. Thus, this statement is true.Therefore, both statements are true.
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