# Simpson's Rule Integrator (∫f(x)dx)

Composite Simpson's 1/3 rule for ∫f(x)dx (even n): node table with weights, integral estimate, optional exact value, O(h⁴) accuracy.

> Canonical page: https://elysiatools.com/en/tools/numerical-integration-simpson

- **Category:** Math & Numbers

- **Keywords:** simpson's rule, simpson rule, numerical integration, integral calculator, approximate integral, quadrature, simpson 1/3, calculus, definite integral, numerical methods

## Overview

The Simpson's Rule Integrator approximates definite integrals using the composite Simpson's 1/3 rule. It evaluates mathematical expressions over an interval divided into an even number of subintervals, producing a node-by-node table with quadrature weights, an integral estimate with O(h⁴) convergence, and optional error analysis against an exact antiderivative.

## Inputs

- **f(x) to integrate** (text): Integrand in x (functions: sin, cos, exp, ln, sqrt, …).
- **Lower Limit a** (number): e.g. 0
- **Upper Limit b** (number): May be less than a (the integral flips sign, as in calculus).
- **Subintervals n (even)** (number): Must be even. h = (b − a)/n; global error is O(h⁴) — halving h divides the error by ~16.
- **Exact Antiderivative F(x) (optional)** (text): Exact antiderivative in x; the tool reports F(b) − F(a) and the absolute error. Leave empty to skip.
- **Decimal Places** (number)

## When to use

- Integrating non-elementary functions like e^(-x^2) or sin(x)/x that lack closed-form antiderivatives.
- Verifying numerical analysis coursework and homework against exact algebraic antiderivatives.
- Approximating definite integrals quickly with high fourth-order accuracy using a modest number of intervals.

## How it works

- Enter the integrand f(x), lower bound a, upper bound b, and an even number of subintervals n to set step size h = (b - a) / n.
- The tool evaluates f(x) at each grid point x_i and applies alternating Simpson weights (1, 4, 2, 4, ..., 1).
- It sums the weighted values and multiplies by h/3 to compute the composite Simpson integral approximation.
- If an exact antiderivative F(x) is supplied, it calculates F(b) - F(a) and outputs the absolute truncation error.

## Use cases

- Evaluating Gaussian probability integrals and error functions over finite intervals.
- Engineering simulations requiring rapid quadrature for non-linear load curves and response spectra.
- Numerical methods coursework demonstrating O(h⁴) error reduction across varying grid sizes.

## Frequently asked questions

### Why does the number of subintervals n need to be an even number?

Composite Simpson's 1/3 rule fits parabolas across pairs of adjacent subintervals, which requires an even number of divisions.

### What functions and operators can I use in f(x)?

You can use standard mathematical operations and built-in functions such as sin, cos, exp, ln, sqrt, and power expressions.

### Why is the estimate exact for quadratic and cubic polynomials?

Simpson's 1/3 rule has a degree of precision of 3, meaning its truncation error term relies on the fourth derivative, which is zero for polynomials up to degree 3.

### Can the upper limit b be smaller than the lower limit a?

Yes. When b is less than a, the step size h is negative, and the integral flips sign according to standard calculus rules.

### What is the purpose of the exact antiderivative field?

Entering F(x) computes the theoretical value F(b) - F(a) to display the exact absolute error alongside the numerical approximation.

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