# Numerical Integration and Differentiation

Quadratures, double and triple integrals, and multidimensional derivatives

Numerical integration functions can approximate the value of an integral whether or not the functional expression is known:

• When you know how to evaluate the function, you can use `integral` to calculate integrals with specified bounds.

• To integrate an array of data where the underlying equation is unknown, you can use `trapz`, which performs trapezoidal integration using the data points to form a series of trapezoids with easily computed areas.

For differentiation, you can differentiate an array of data using `gradient`, which uses a finite difference formula to calculate numerical derivatives. To calculate derivatives of functional expressions, you must use the Symbolic Math Toolbox™ .

## Functions

expand all

 `integral` Numerical integration `integral2` Numerically evaluate double integral `integral3` Numerically evaluate triple integral `quadgk` Numerically evaluate integral — Gauss-Kronrod quadrature `quad2d` Numerically evaluate double integral — tiled method
 `cumtrapz` Cumulative trapezoidal numerical integration `trapz` Trapezoidal numerical integration
 `del2` Discrete Laplacian `diff` Differences and approximate derivatives `gradient` Numerical gradient
 `polyint` Polynomial integration `polyder` Polynomial differentiation

## Topics

Integration to Find Arc Length

This example shows how to parametrize a curve and compute the arc length using `integral`.

Complex Line Integrals

This example shows how to calculate complex line integrals using the `'Waypoints'` option of the `integral` function.

Singularity on Interior of Integration Domain

This example shows how to split the integration domain to place a singularity on the boundary.

Analytic Solution to Integral of Polynomial

This example shows how to use the `polyint` function to integrate polynomial expressions analytically.

Integration of Numeric Data

This example shows how to integrate a set of discrete velocity data numerically to approximate the distance traveled.

Calculate Tangent Plane to Surface

This example shows how to approximate gradients of a function by finite differences.