GATE AG 2007 • Q5
MCQ • 1 Mark
Section 1: Engineering Mathematics
Numerical Methods
Subtopic: Numerical integration – trapezoidal and Simpson's rule
Question Statement:
Integrating the function $f(x) = 1 + e^{-x}\sin(4x)$ over the interval $[0, 1]$ using Simpson's $1/3$rd rule gives:
Options:
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(A) 1.021
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(B) 1.091
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(C) 1.321
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(D) 2.642
Step-by-Step Solution & Explanation
Simpson's 1/3 rule with n=2 (h=0.5): x₀=0,x₁=0.5,x₂=1. f(0)=1, f(0.5)=1+e^{−0.5}sin(2)=1+0.6065×0.9093=1.5515, f(1)=1+e^{−1}sin(4)=1+0.3679×(−0.7568)=0.7216. I = (h/3)[f₀+4f₁+f₂] = (0.5/3)[1+6.206+0.7216] = 1.321. (A) underestimates using trapezoidal-like averaging; (D) results from omitting the 1/3 factor.
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