Paths, Arc Length in Polar Coordinates

Arc Length in Polar Coordinates

In an earlier section we computed the velocity of a path in polar coordinates as r′,rθ′. Convert this to speed, giving sqrt(r′² + r²θ′²). Integrate this expression to get arc length.

Sometimes r is given as a function of θ, just as y was a function of x in the earlier "parabola" example. In this case θ(t) = t, and the integrand simplifies to sqrt(r²+r′²).