Arc length in polar curves

Goal : find a formula for the arc length of a curve in polar coordinates

Arc length of a curve in polar coordinates

To find the formula for the arc length of a curve in polar coordinates, let's start from the formula of the arc length of a parametrized curve (x(t), y(t)) for a≤ t ≤b  in rectangular coordinates.

In polar coordinates the curve is defined by r = f(θ) and we also have:

x = rcosθ = f(θ)cosθ and y = rsinθ =f(θ)sinθ . Let's calculate dx/dθ and dy/dθ:

Let's replace dt by dθ and a and b by ɑ and β, which define the limits of integration of the curve in polar coordinates, in the formula for the arc length above:

This leads to the following theorem:

Theorem