GATE AG 2012 • Q26 NAT • 1 Mark

Section 1: Engineering Mathematics

Numerical Methods

Subtopic: Numerical integration – trapezoidal and Simpson's rule

Question Statement:

The value of $\int_{0}^{\pi/2} \cos x \, dx$ using the trapezoidal rule with two equal intervals is:

Official Answer Key:

0.94 to 0.96

Step-by-Step Solution & Explanation

h = (π/2)/2 = π/4. Nodes: x0=0 (cos=1), x1=π/4 (cos=0.7071), x2=π/2 (cos=0). Trapezoidal rule: I = (h/2)[f0 + 2f1 + f2] = (π/8)[1 + 1.4142 + 0] = 0.3927 × 2.4142 = 0.948, within the accepted 0.94–0.96 range. (Exact value = 1, so the 2-interval trapezoidal rule under-estimates by ≈5%, typical of trapezoidal error for a concave function.)

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