Simpson's Rule
Numerical integration method approximating definite integrals. Uses parabolas instead of straight lines. Formula: (Δx/3)[f(x₀) + 4f(x₁) + 2f(x₂) + ... + f(xₙ)].
Formula
Coefficients: 1, 4, 2, 4, 2, ..., 4, 1. Works for complex functions without simple antiderivatives.
Steps
1) Even number of intervals. 2) Calculate Δx. 3) Generate x-values. 4) Evaluate function. 5) Apply coefficients. 6) Multiply by Δx/3.
Example
eˣ from 0 to 2 with n=4. Δx=0.5. Result ≈ 6.391 (exact: 6.389).
Accuracy
Typically < 0.1% error. Reduces by factor 16 when doubling n.
Applications
Engineering, Physics, Economics, Medicine, Environmental science.
Conclusion
Master numerical integration for complex functions.