What is Simpson's rule?
A numerical integration method fitting parabolas through consecutive triples of points.
Simpson's Rule Calculator
Use parabolic arcs on pairs of subintervals—n must be even.
Set up Simpson’s rule
Choose f(x)=x² with even n, or provide an odd count of y-values (n even).
About this calculator
How to use Simpson's Rule Calculator, plus the formulas and logic behind the results.
Tool description
Simpson's 1/3 rule weights samples 1-4-2-4-…-4-1 and is exact for cubics (hence for x²).
How to use
- Select x² mode or y-list mode.
- Ensure n is even (y-list length must be odd).
- Select Calculate to view the approximation.
- Compare with the exact integral when known.
Formulas & reference
S = (h/3)[y₀ + 4y₁ + 2y₂ + 4y₃ + … + yₙ]h = (b − a)/n
Simpson tips
- n even ⇒ number of points = n+1 is odd
- More accurate than trapezoidal for smooth f
- Exact for f(x)=x² in theory
- h must match the spacing of your y list
Calculation logic
Applies composite Simpson weights to equally spaced samples; rejects odd n.
Requires even n. For polynomials of degree ≤ 3 the rule is exact (in exact arithmetic).
Simpson's Rule Calculator FAQ
Common questions about this calculator and how to use the results.
Why even n?
Each parabolic piece covers two subintervals.
Y-list length?
Must be odd: y₀…yₙ with n even.
Is it exact for x²?
Yes (up to floating-point error).
Trapezoid vs Simpson?
Simpson generally has a higher order of accuracy.
Is this free?
Yes.