Definite Integral Calculator

Approximate a finite real expression integral with 256-panel composite Simpson quadrature; the refinement error estimate assumes a smooth integrand and is not a certified bound or improper integral solver.

Description

Approximate a finite real integral with 256-panel Simpson quadrature and a refinement error estimate.

A definite integral can be approximated by weighted samples across a finite interval. This tool uses fixed composite Simpson quadrature and compares two resolutions to report a heuristic error estimate, rather than finding a symbolic primitive.

When to use this tool

Approximate a finite real integral with 256-panel Simpson quadrature and a refinement error estimate. Use it when the input measurements and conventions below match the operation you need.

Prepare expression, variable, lowerBound, upperBound, respecting the measurement basis and boundaries below.
Compare the worked example with your intended operation before reusing its result.

How to use the tool and how it works

Enter expression with explicit multiplication, choose variable or keep its x default, and supply finite integration bounds if shown.
Run the tool. Correct input errors before interpreting an output.
Read the named output fields below. Check the symbolic domain or the numerical-method limitations before using the answer.

Enter a supported real expression, a case-sensitive variable defaulting to x and finite bounds. The tool checks evaluation at the lower bound, samples 257 equally spaced points across ordered bounds and applies Simpson weights for 256 panels. Every second sample gives a 128-panel result. integral uses the fine result with reversed-bound sign, estimatedError is abs(fine minus coarse)/15, and method is composite-simpson-256. Equal valid bounds return zero integral and error.^[1]

Input field	Meaning and units	Accepted input
expression	Real expression with explicit multiplication and one free variable, e.g. x^2 or sin(x).	Nonempty expression text, at most 4,096 characters; explicit multiplication
variable	Case-sensitive integration variable; other unbound symbols are rejected.	Text; optional, default "x"
lowerBound	Lower bound.	Finite number
upperBound	Upper bound.	Finite number

Output field	Interpretation
integral	Signed 256-panel composite Simpson approximation; reversing bounds reverses its sign.
estimatedError	Absolute difference of 256- and 128-panel results divided by 15; heuristic under smoothness assumptions, not a guaranteed bound.
method	The fixed method identifier composite-simpson-256.

json

{
  "input": {
    "expression": "x^2",
    "variable": "x",
    "lowerBound": 0,
    "upperBound": 3
  },
  "output": {
    "integral": 9,
    "estimatedError": 0,
    "method": "composite-simpson-256"
  }
}

Limitations and assumptions

The error estimate assumes adequate smoothness and is not a certified bound. Sampling can miss a narrow feature or singularity between points. The tool is not adaptive and does not solve improper integrals; nonfinite evaluations, unknown free symbols and invalid domains are rejected.
The expression must use explicit multiplication, such as 2*x, and supported function calls. Variable names are case-sensitive ASCII letters followed by letters or digits; constants and function names are reserved. Only variable is optional, defaulting to x.
Expressions are limited to 4,096 characters and bounded nesting and operation counts. Constant-domain errors are rejected; symbolic results do not certify every variable-dependent domain constraint.

Alternative or Complementary methods

Use symbolic differentiation to check a proposed primitive where supported, or numerical quadrature to compare values over a regular finite interval. Sample-based integration requires its own smoothness and domain checks. Compare the scope of the nearby tools below before substituting one.

References

Tool behavior and example: Definite Integral Calculator — tool.io

Similar or alternative tools

simpson integration
Approximate a definite integral from evenly spaced samples over [xMin, xMax] using composite Simpson's rule (odd sample count required).
Indefinite Integral Calculator
Find a symbolic antiderivative for sums, monomials, constant factors and supported elementary functions of linear expressions, with a constant of integration.