Arc Length of Polar Curves - Calculus 2

Another application of calculus within the polar coordinate system is the ability to calculate arc length of polar curves via integration. In the past we already saw how to calculate arc length of curves in the rectangular coordinate system in both rectangular form and parametric form. In fact, we can use the formula for calculating arc length in parametric form to help us develop a nice formula for calculating arc length in polar form.

Since a polar equation can be rewritten as a pair of parametric equations with a parameter θ, we can use this to our advantage in calculating arc length. More specifically, what we can do is represent a polar curve, r=f(θ), with two parametric equations using x=rcosθ and y=rsinθ, which are two of the conversion formulas we used for converting between polar coordinates and rectangular coordinates. We use those two formulas in particular because they relate x and y individually to r and θ. Then after replacing r with f(θ) in these formulas, we create a set of parametric equations that represent the polar curve where the parameter is θ, as seen below.

x=f(θ)cosθ , y=f(θ)sinθ 

These parametric equations can be used in the formula to calculate arc length of parametric curves, which will ultimately allow us to calculate the arc length of polar curves. This involves taking the derivative of each parametric equation with respect to θ, and using those derivatives in the formula. Once simplified, we find a very nice formula for calculating arc length in polar form that can be used for any polar curve that we might want to know the arc length of between two angles of θ.

We discuss all of this and more in this new lesson of Calculus 2. Let's get to it!

In this lesson, you will learn:

• The derivation of the formula to calculate arc length in polar form
• How to use the formula and solve integrals to find the arc length of polar curves

I hope you find this video to be helpful!

-Josh