GATE AG 2019 • Q26
NAT • 1 Mark
Section 1: Engineering Mathematics
Numerical Methods
Subtopic: Numerical integration – trapezoidal and Simpson's rule
Question Statement:
Using the trapezoidal rule, the value of the integral $I = \int_{4.0}^{5.2} \ln(x) \, dx$ (rounded to three decimal places) is ______
Given values:
| $x$ | 4.0 | 4.2 | 4.4 | 4.6 | 4.8 | 5.0 | 5.2 |
|---|---|---|---|---|---|---|---|
| $y = \ln(x)$ | 1.386 | 1.435 | 1.482 | 1.526 | 1.569 | 1.609 | 1.648 |
Official Answer Key:
1.820 to 1.830
Step-by-Step Solution & Explanation
Trapezoidal rule with h = 0.2, n = 6 intervals:
I = (h/2)[y₀ + y₆ + 2(y₁+y₂+y₃+y₄+y₅)]
= (0.2/2)[1.386 + 1.648 + 2(1.435+1.482+1.526+1.569+1.609)]
= 0.1 × [3.034 + 2(7.621)]
= 0.1 × [3.034 + 15.242] = 0.1 × 18.276 = 1.8276 ≈ 1.828. Falls within the official range 1.820–1.830.
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