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™
.

**Integration to Find Arc Length**

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

.

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.

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.