Numerical Integration Calculator

Approximate integrals with the trapezoidal, midpoint and Simpson’s rules and compare them with an adaptive reference value.

Your result will appear here

Fill in the fields and press Calculate, or pick one of the examples.

About the Numerical Integration Calculator

Approximates a definite integral with the classical composite rules — trapezoidal, midpoint and Simpson’s — for a number of subintervals you choose, and compares each with a high-accuracy adaptive reference value.

It is a tool for studying the methods themselves: double n and watch the trapezoidal error fall by about 4 and Simpson’s by about 16.

Limitations

Finite limits only, and the integrand must be finite at every sample point including the endpoints.

How to use it

Type the integrand and the limits.
Choose the number of subintervals n (even for Simpson’s rule).
Press Calculate and compare the methods in the table.

Example

∫₀¹ x² dx, n = 4

→ Trapezoid 0.34375, Simpson 0.333333333333

Formulas used

T_n = \frac{h}{2}\left[f(x_0) + 2f(x_1) + \dots + 2f(x_{n-1}) + f(x_n)\right]

Trapezoidal rule, error of order $h^2$ .

S_n = \frac{h}{3}\left[f(x_0) + 4f(x_1) + 2f(x_2) + \dots + 4f(x_{n-1}) + f(x_n)\right]

Simpson’s rule (n even), error of order $h^4$ .

Definite Integral CalculatorEvaluate definite and improper integrals: analytically when an antiderivative exists, otherwise by adaptive quadrature.Numerical Differentiation CalculatorApproximate derivatives with forward, backward, central and five-point differences and compare with the symbolic value.

About the Numerical Integration Calculator

Limitations

How to use it

Formulas used

Related tools

About the Numerical Integration Calculator

Limitations

How to use it

Formulas used