Simpson’s Rule Calculator
Approximate a definite integral from a mathematical function or sampled data, verify interval requirements, and compare Simpson’s result with the trapezoidal rule.
Simpson’s Rule Calculator inputs
Integrate a supported mathematical expression with an even number of subintervals, or paste ordered x,y samples. A separate trapezoidal estimate is always shown.
Simpson’s Rule Calculator results
How to use the Simpson’s rule calculator
The Simpson’s rule calculator uses a parser that accepts defined mathematical functions without executing arbitrary code. Composite one-third weighting is applied only when samples are equally spaced and the interval count is even.
Composite Simpson one-third rule
The Simpson’s rule calculator groups adjacent pairs of equal-width subintervals and replaces the function locally with a quadratic interpolant. Endpoints receive weight one, odd interior samples weight four and even interior samples weight two. The number of subintervals must therefore be even for the standard composite one-third rule.
Safe function parser
Function mode uses a dedicated mathematical tokenizer and parser rather than executing entered JavaScript. It supports x, parentheses, arithmetic operators, powers, pi, e and common functions such as sin, cos, tan, exp, log, sqrt and abs. Unsupported names or malformed expressions produce a validation message instead of running code.
Tabulated data requirements
Data mode expects ordered x,y pairs. Composite Simpson’s one-third rule requires equal spacing and an odd number of sample points. When those conditions are not met, the calculator reports why Simpson’s rule is unavailable while still showing a trapezoidal estimate for the valid ordered data.
Accuracy and error interpretation
Simpson’s rule is exact for polynomials through degree three when the sampling assumptions hold, but accuracy still depends on smoothness and interval width. Sharp peaks, discontinuities, singularities and oscillations may require interval splitting or adaptive methods. Comparing Simpson and trapezoidal estimates is useful, but their difference is not a guaranteed error bound.
Simpson’s Rule Calculator questions
Why must the number of subintervals be even?
Simpson’s one-third rule fits one parabola across each pair of subintervals, so the composite method needs an even count.
Can I enter degrees in sin or cos?
Trigonometric functions use radians. Convert degrees to radians inside the expression, for example sin(x*pi/180).
What happens with uneven x spacing?
The calculator declines the standard equal-spacing Simpson formula and reports a trapezoidal result instead.