## 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.

`$\underset{0}{\overset{3\pi }{\int }}\sqrt{4{\mathrm{cos}}^{2}\left(2t\right)+{\mathrm{sin}}^{2}\left(t\right)+1}\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}dt.$`

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`.