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 Symbolic Math Toolbox™.

Functions

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Integrate Functional Expressions

`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

Integrate Numeric Data

`cumtrapz`	Cumulative trapezoidal numerical integration
`trapz`	Trapezoidal numerical integration

Finite Difference Derivatives

`del2`	Discrete Laplacian
`diff`	Differences and approximate derivatives
`gradient`	Numerical gradient

Polynomials

`polyint`	Polynomial integration
`polyder`	Polynomial differentiation

Topics

Integration to Find Arc Length
This example shows how to parameterize 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.

Related Information

Numerical Methods with Applications (MathWorks Teaching Resources)