Allied & ICAR Exam Bank • ALLIED_1006
NAT • 2 Marks
Section 1: Engineering Mathematics
Numerical Methods
Subtopic: Numerical integration
Question Statement:
Evaluating $\int_0^2 x^3 dx$ using Simpson's $1/3$ rule with a step size of $h = 1$ yields an estimated value of:
Step-by-Step Solution & Explanation
With $h = 1$, the grid points are $x_0 = 0, x_1 = 1, x_2 = 2$. Function values are $f(0) = 0, f(1) = 1, f(2) = 8$. By Simpson's $1/3$ rule: $I \approx \frac{h}{3}[f(x_0) + 4f(x_1) + f(x_2)] = \frac{1}{3}[0 + 4(1) + 8] = \frac{12}{3} = 4.0$. (Note: Simpson's rule is exact for polynomials of degree up to 3).
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