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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(2*t*), *y*(*t*) = cos(*t*), *z*(*t*) = *t*,

where *t* ∊ [0,3*π*].

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.

$$\underset{0}{\overset{3\pi}{\int}}\sqrt{4\mathrm{cos}{(2t)}^{2}+\mathrm{sin}{(t)}^{2}+1}}\text{\hspace{0.05em}}dt.$$

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`

.