Question Bank • QB_EM_NM_005
NAT • 2 Marks
Section 1: Engineering Mathematics
Numerical Methods
Subtopic: Numerical integration – trapezoidal and Simpson's rule
Question Statement:
Evaluating $\int_0^1 \frac{1}{1 + x} \, dx$ using the Trapezoidal rule with a step size $h = 0.5$ yields ________ (round off to 2 decimal places).
Step-by-Step Solution & Explanation
The interval is $[0, 1]$ with $h = 0.5$.
The nodes are $x_0 = 0, x_1 = 0.5, x_2 = 1.0$.
Compute the ordinates:
$y_0 = f(0) = \frac{1}{1 + 0} = 1$
$y_1 = f(0.5) = \frac{1}{1 + 0.5} = \frac{1}{1.5} = \frac{2}{3} \approx 0.6667$
$y_2 = f(1.0) = \frac{1}{1 + 1} = \frac{1}{2} = 0.5$
By the Trapezoidal rule:
$I = \frac{h}{2} [y_0 + 2y_1 + y_2] = \frac{0.5}{2} \left[ 1 + 2\left(\frac{2}{3}\right) + 0.5 \right]$
$I = 0.25 \left[ 1.5 + \frac{4}{3} \right] = 0.25 \left[ \frac{9 + 8}{6} \right] = 0.25 \times \frac{17}{6} = \frac{4.25}{6} \approx 0.7083$
Rounding off to 2 decimal places gives $0.71$.
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