Numerical Methods

Given interval $[a, b] = [0, 2]$ and $h = 1$. The grid points are $x_0 = 0, x_1 = 1, x_2 = 2$. Function values: $y_0 = f(0) = 0^3 = 0$ $y_1 = f(1) = 1^3 = 1$ $y_2 = f(2) = 2^3 = 8$ Simpson's $1/3$ rule formula: $I = \frac{h}{3} [y_0 + 4y_1 + y_2] = \frac{1}{3} [0 + 4(1) + 8] = \frac{12}{3} = 4$ (Note: Simpson's $1/3$ rule is exact for polynomials up to degree 3, and the exact integral is $\left[\frac{x^4}{4}\right]_0^2 = 4$).