# Documentation

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## Integration to Find Arc Length

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

Consider the curve parameterized by the equations

x(t) = sin(2t),  y(t) = cos(t),  z(t) = t,

where t ∊ [0,3π].

Create a three-dimensional plot of this curve.

```t = 0:0.1:3*pi; plot3(sin(2*t),cos(t),t) ```

The arc length formula says the length of the curve is the integral of the norm of the derivatives of the parameterized equations.

`$\underset{0}{\overset{3\pi }{\int }}\sqrt{4\mathrm{cos}{\left(2t\right)}^{2}+\mathrm{sin}{\left(t\right)}^{2}+1}\text{ }dt.$`

Define the integrand as an anonymous function.

`f = @(t) sqrt(4*cos(2*t).^2 + sin(t).^2 + 1);`

Integrate this function with a call to `integral`.

`len = integral(f,0,3*pi)`
```len = 17.2220 ```

The length of this curve is about `17.2`.