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QB_EM_ADV_081
Question Bank • QB_EM_ADV_081
NAT • 2 Marks
Section 1: Engineering Mathematics
Numerical Methods
Subtopic: Numerical integration – trapezoidal and Simpson's rule
Question Statement:
Evaluate the definite integral $\int_0^1 \frac{1}{1 + x} \, dx$ using the trapezoidal rule with $n = 2$ equal subintervals (step size $h = 0.5$). The calculated numerical value is ________ (round off to three decimal places).
Step-by-Step Solution & Explanation
Step size $h = \frac{1 - 0}{2} = 0.5$.
The evaluation nodes and functional values $y = f(x) = \frac{1}{1 + x}$ are:
- $x_0 = 0.0 \implies y_0 = \frac{1}{1 + 0} = 1.000$
- $x_1 = 0.5 \implies y_1 = \frac{1}{1 + 0.5} = \frac{2}{3} \approx 0.6667$
- $x_2 = 1.0 \implies y_2 = \frac{1}{1 + 1} = 0.500$
Applying the composite trapezoidal formula for $n = 2$:
$I = \frac{h}{2} [y_0 + 2y_1 + y_2] = \frac{0.5}{2} \left[ 1.0 + 2\left(\frac{2}{3}\right) + 0.5 \right]$
$= 0.25 \left[ 1.5 + \frac{4}{3} \right] = 0.25 \left[ \frac{9 + 8}{6} \right] = 0.25 \times \frac{17}{6} = \frac{17}{24} \approx 0.7083 \approx 0.708$
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