Arc Length with Polar Coordinates

Here we need to move into the applications of integrals and how we do them in terms of polar coordinates.  In this part we will look at the arc length of the curve specified by,

r = f (θ)

α < θ < β

where we as well assume that the curve is traced out precisely once. Just only as we did with the tangent lines in polar coordinates we will very first write the curve in terms of a set of parametric equations,

 x = r cosθ

= f (θ) cosθ

 y = r sinθ

= f (θ) sin θ

and we can now make use of the parametric formula for finding the arc length.

We'll require the following derivatives for these computations.

dx/ d θ = f' (θ) cosθ - f (θ) sinθ

= dr/dθ cosθ - r sinθ

 dy/dθ = f' (θ) sinθ + f (θ) cosθ

= dr/dθ sinθ + r cosθ

We'll require the following for our ds.

After that the arc length formula for polar coordinates is,

Where

L = ∫ ds

ds = √(r² + (dr/dθ)²) dθ