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QB_EM_ADV_007
Question Bank • QB_EM_ADV_007
NAT • 2 Marks
Section 1: Engineering Mathematics
Numerical Methods
Subtopic: Numerical integration – trapezoidal and Simpson's rule
Question Statement:
The integral $\int_{0}^{6} \frac{1}{1 + x^2} \, dx$ is evaluated using Simpson's $3/8$ rule with $n = 6$ subintervals (step size $h = 1$). The calculated value of the integral is ________ (round off to two decimal places).
Step-by-Step Solution & Explanation
Simpson's $3/8$ rule for $n = 6$ subintervals of step $h = 1$:
$I = \frac{3h}{8} \left[ y_0 + y_6 + 3(y_1 + y_2 + y_4 + y_5) + 2(y_3) \right]$
Evaluating $y = f(x) = \frac{1}{1 + x^2}$ at grid points:
- $y_0 = f(0) = 1.0000$
- $y_1 = f(1) = 0.5000$
- $y_2 = f(2) = 0.2000$
- $y_3 = f(3) = 0.1000$
- $y_4 = f(4) = 0.0588$
- $y_5 = f(5) = 0.0385$
- $y_6 = f(6) = 0.0270$
Sum terms:
- $y_0 + y_6 = 1.0000 + 0.0270 = 1.0270$
- $3(y_1 + y_2 + y_4 + y_5) = 3(0.5000 + 0.2000 + 0.0588 + 0.0385) = 3(0.7973) = 2.3919$
- $2(y_3) = 2(0.1000) = 0.2000$
Total sum inside brackets $= 1.0270 + 2.3919 + 0.2000 = 3.6189$.
$I = \frac{3(1)}{8} \times 3.6189 = 0.375 \times 3.6189 = 1.3571 \approx 1.36$
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