Simpson’s Rule for Numerical Integration

← Numerical Methods

Source inspiration: (Mathew 2000-2019).

Animations

Each animation below shows the Simpson’s rule diagram building up parabolic panel by panel over \([0, 2]\) for \(y = e^{-x}\sin(8x^{2/3})+1\). Each frame adds one new parabolic panel (spanning 2 subintervals); the running sum is annotated. Simpson’s rule fits a parabola through 3 consecutive nodes — achieving \(O(h^4)\) accuracy.

Julia source scripts that generated these animations are linked under each case.

Case 1 — Simpson’s Rule, \(y = e^{-x}\sin(8x^{2/3})+1\), \(n=10\) subintervals

Behavior: Each panel uses 3 nodes \((x_{2i}, x_{2i+1}, x_{2i+2})\) to fit a parabola. The area under each parabola is \(\frac{h}{3}(f_0 + 4f_1 + f_2)\). The error is \(O(h^4)\), far better than the trapezoidal rule.

Julia source

Simpson’s Rule animation: 5 parabolic panels build one-by-one over [0,2]; each panel spans 2 sub-intervals with a parabolic top; running sum annotated each frame

Derivation Notes (Planned)

Short derivations will be added to explain the core equations and assumptions.

Worked Example (Planned)

A compact numerical example with intermediate steps will be included.

Implementation Notes (Planned)

Implementation details, numerical stability notes, and practical pitfalls will be added.

Legacy Animation Inventory (Stub)

Animation Portals

  1. Simpson’s Rule (ok)
  • Animation item links found: 1

Animation Items

  1. Simpson’s Rule (ok)
  • Main animated GIF count: 1
  • http://localhost:8000/a2001/Animations/Quadrature/Simpson/Simpsonaa.gif

References

Mathew, John H. 2000-2019. Numerical Analysis - Numerical Methods Modules. https://web.archive.org/web/20190808102217/http://mathfaculty.fullerton.edu/mathews/n2003/NumericalUndergradMod.html.