Mathematics / Free calculator

Numerical integration

Approximate a definite integral with midpoint, trapezoid, Simpson 1/3, or Simpson 3/8 quadrature.

01 Your inputs

Use x, + − * / ^, sin, cos, exp, ln, sqrt. Angles are in radians. Use explicit multiplication for clarity.

Entered function
f(x)=x3
Quadrature rule

Calculated on your device. Inputs are not transmitted or stored.

02 Your result

Enter your values and select Calculate.

The method

∫abf(x)dx≈h∑wif(xi)

A worked example

For ∫₀¹ x³ dx = 1/4, both Simpson rules return 0.25 with 12 subintervals. Reverse the bounds to get −0.25.

Assumptions & limits

1–120 whole subintervals. Simpson 1/3 requires even n; Simpson 3/8 requires n divisible by 3. Equal bounds give zero for a defined integrand. Function evaluations must be finite; discontinuities and integrable singularities are not detected automatically. Repeat with larger n to assess convergence. In the schematic sum, weights include each rule’s normalization. Full double precision between operations; displayed values use up to 10 decimal places. Numerical outputs above 10¹⁰⁰ and nonfinite function evaluations are rejected.

Calculation source: Math Workbench: composite midpoint, trapezoid, and Simpson rules.

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