Evaluate the definite integral $I = \int_{0}^{4} x^2 \, dx$ numerically using Simpson's $1/3$ rule with a step size $h = 1$ (4 subintervals). What is the evaluated numerical value?

Answer Key:

21.33

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Quadrature Rule:** For $h = 1$, grid points are $x = 0, 1, 2, 3, 4$ with $f(x) = 0, 1, 4, 9, 16$. 2. **Stage 2: Simpson Formula:** $I = \frac{h}{3} [f_0 + 4(f_1 + f_3) + 2(f_2) + f_4] = \frac{1}{3} [0 + 4(1 + 9) + 2(4) + 16] = \frac{1}{3}[40 + 8 + 16] = \frac{64}{3} \approx 21.33$. 3. **Stage 3: Precision Check:** Because $f(x)$ is a quadratic polynomial, Simpson's 1/3 rule yields exact mathematical value. 4. **Stage 4: Accepted Range:** 21.30 to 21.36.

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