GATE 2020 AE – Question 46

Engineering Mathematics · Numerical Methods: Bisection and Newton-Raphson, numerical differentiation, trapezoidal and Simpson's rules · 2 marks · Numerical answer

$\int_0^1(x^2-2x+1)dx$ is evaluated by the trapezoidal rule with four intervals. Find numerical minus analytical integral, to 3 decimals.

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Correct answer: 0.010 to 0.012

Explanation

The exact integral is 1/3. For h = 1/4, ordinates are 1, 9/16, 1/4, 1/16 and 0. The trapezoidal value is $(1/8)[1+2(9/16+1/4+1/16)]=11/32=0.34375$. Difference is $11/32-1/3=1/96=0.0104167$, approximately 0.010. Convexity explains the positive trapezoidal error.

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