Simpson's Rule

Numerical Analysis · Axiom Academy

Approximating integrals with parabolas for higher accuracy Instead of connecting endpoints with a straight line (trapezoid), Simpson's Rule fits a parabola through three points: the two endpoints and the midpoint. Simpson's Rule uses three function evaluations with weights 1, 4, 1: Where m = (a+b)/2 is the midpoint. The coefficient pattern 1-4-1 comes from integrating the Lagrange interpolating polynomial. 3. Where Do the Coefficients Come From? The weights come from integrating the Lagrange interpolating polynomial through three points: Let's approximate the same integral as before: ∫₀¹ x² dx The error in Simpson's Rule depends on the fourth derivative:

This is the written version of the interactive lesson above. See the full Numerical Analysis course.