Use this numerical method to approximate a derivative or definite integral from a callable expression and explicit step or interval settings.

Samples

Function samples at evenly spaced points, first at xMin and last at xMax; the count must be odd.

xMin

Lower integration bound.

xMax

Upper integration bound.

Approximate a definite integral from evenly spaced samples over [xMin, xMax] using composite Simpson's rule (odd sample count required). The tool evaluates the supplied inputs together and returns the named outputs below; it does not infer omitted operating conditions or change the units shown.^[1]

Integral

Simpson approximation of the integral.

Delta x

Sample spacing implied by the bounds.

• Numerical differentiation and integration introduce truncation and floating-point error. Discontinuities, singularities, oscillation, cancellation, or an unsuitable step size can make a plausible-looking result inaccurate.
• Use finite inputs in the displayed units and preserve more precision than the final presentation requires. Independently verify safety-critical, financial, compliance, or production decisions.

Repeat the calculation with a finer resolution and compare the change. Use symbolic calculus or an adaptive high-precision method when an exact result or controlled error bound is required.

• Trapezoidal Integration Calculator

  Approximate a definite integral from evenly spaced samples over [xMin, xMax] using the composite trapezoidal rule.

• Numerical Derivative

  Approximate first and second derivatives of a polynomial via central finite differences.

• Rarefied Species Curve Point

  Estimate a rarefied species count with a documented half-saturation form: species total times sample over sample plus half-saturation, where the half-saturation defaults to the total individual count rounded up to its half.