Arc Length of Parametric Curves - Calculus 2

Earlier in Calculus 2 (Lesson 6), we saw how to use a definite integral to calculate the arc length of a curve between two values of x. Up until now, we always calculated this arc length by working in terms of x because those curves had always been defined with functions in terms of x and y, or rectangular equations. However, now we will look at how to calculate the arc length for plane curve/parametric curves represented with parametric equations.

To calculate the arc length in terms of the parameter (typically t), we need to first find the arc length formula in terms of the parameter. This is done by using the parametric equations that represent the curve to convert the original arc length formula from being in terms of x, to being in terms of the parameter, t. Once found, this formula can be applied to any scenario where we want to calculate the arc length of a parametric curve (provided that the curve is a smooth curve without any gaps, sharp points, or points where the curve intersects itself). 

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:

• Where the formula for arc length of parametric curves originates (the derivation)
• How to set up an integral in terms of the parameter to represent arc length of parametric curves
• How to solve integrals in terms of the parameter to calculate arc length

I hope you find this video to be helpful!

-Josh