The line integral along the curve C can be written as

where C is parametrisized as r(t) with the parmeter t.

∫¦r'(t)¦ dt equals the arc length i.e. the length of the curve.

Consider eg the circle.  x²+ y² = r². This can be parametrisized as follows

x(t) = cos(t)

y(t) = sin(t)

The perimeter of the circle can then be calculated according to

2π

∫ sint²+cos²t dt= 2π
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