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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.

03π4cos(2t)2+sin(t)2+1dt.

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.

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