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Calculus · Numerical integration

Definite Integral Approximation Calculator

Approximate a polynomial’s definite integral with composite trapezoidal or Simpson’s rule and compare it with the exact polynomial antiderivative.

Formula shown Runs locally Reviewed Aug 11, 2026
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Equation guide

Definite integration of a polynomial using a uniform partition

Step width

h = (b−a)/n

Trapezoidal rule

Tₙ = h[f(a)/2 + Σf(xᵢ) + f(b)/2]

Simpson’s rule

Sₙ = h[f₀+4f₁+2f₂+…+fₙ]/3
Calculus · Numerical methods

Approximate a definite integral

Inputs and displayed decimal results use at most three decimal places. Exact internal values are retained during calculation.

Define a polynomial, bounds, method, and subdivision count.

Using the model

Symbols, assumptions and limitations

Interpretation notes

The integrand is a real polynomial of degree one through three. Bounds must differ, and Simpson’s rule requires an even number of subintervals. The exact polynomial integral is shown only as an educational error check.

Worked examples

See the method in practice

01

Integral of x²

On [0,1], Simpson’s rule reproduces the exact value 1/3 for x² because it is exact for polynomials through degree three.

02

Trapezoidal refinement

Increasing the subinterval count reduces the straight-line approximation error for a smooth curved integrand.

Questions

Frequently asked

Why must Simpson’s n be even?

Composite Simpson’s rule fits a quadratic across pairs of adjacent subintervals.

Why show an exact value?

For supported polynomials an antiderivative is available, allowing the numerical method and its error to be checked.

Does this integrate arbitrary expressions?

No. The first reviewed version uses polynomial coefficients and avoids an unrestricted expression parser.

Sources & review

Equations and examples are checked against the references below. Results are educational and should be independently verified for safety-critical work.

OpenStax Calculus Volume 2 — Numerical Integration

Written by the STEM Hub editorial team · Reviewed August 11, 2026 · Review methodology