Integration to Find Arc Length

This example shows how to parameterize 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.

$\int_{0}^{3 π} \sqrt{4 \cos^{2} (2 t) + \sin^{2} (t) + 1} d t .$

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.

See Also