Question Bank • QB_EM_ADV_082 NAT • 2 Marks

Section 1: Engineering Mathematics

Numerical Methods

Subtopic: Numerical integration – trapezoidal and Simpson's rule

Question Statement:

Evaluate the integral $\int_0^2 e^{x^2} \, dx$ using Simpson's $1/3$ rule with $n = 2$ subintervals ($h = 1$). The calculated numerical value is ________ (round off to two decimal places).

Answer Key:

22.16

Step-by-Step Solution & Explanation

**Method 1: Simpson's 1/3 Rule Formula** For $n = 2$ subintervals over $[0, 2]$, step size $h = 1$: $I = \frac{h}{3} [y_0 + 4y_1 + y_2]$ Evaluating the integrand $f(x) = e^{x^2}$ at grid nodes: - $y_0 = f(0) = e^0 = 1.0000$ - $y_1 = f(1) = e^1 \approx 2.71828$ - $y_2 = f(2) = e^4 \approx 54.59815$ Substitute into the formula: $y_0 + 4y_1 + y_2 = 1.0000 + 4(2.71828) + 54.59815 = 1.0000 + 10.87312 + 54.59815 = 66.47127$ $I = \frac{1}{3} \times 66.47127 = 22.1571 \approx 22.16$
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