To find the area under a curve, slice it into strips — but instead of topping each strip with a flat line (crude) or a slanted one (trapezoid), Simpson’s rule fits a little PARABOLA over each pair of strips. Curves hug parabolas well, so the error plummets: a handful of strips already nails the integral. It’s how numerical integration is done. Slide to add strips and watch the estimate lock onto the true area.
Simpson’s rule approximates ∫ᵏᵇf by fitting a PARABOLA through each consecutive triple of points: ∫ ≈ (h/3)[f₀+4f₁+2f₂+4f₃+…+fₙ]. Because a quadratic fit cancels more error than flat (midpoint) or linear (trapezoid) pieces, its error is O(h⁴) — and it integrates any cubic EXACTLY. So ∫₀³x² = 9 is recovered to machine precision with few strips. It (and adaptive/Gauss quadrature) is the standard for definite integrals with no closed form. A fail-loud self-check throws unless ∫₀³x² evaluates to 9 within 1e−6. ◆ real numerical methods, node-verified.
Simpson needs an even number of intervals and a smooth integrand; on sharp/oscillatory or singular functions it errs (adaptive or Gauss quadrature help). The O(h⁴) accuracy and exact-on-cubics property are exact.