Arc Length of a Curve!

The arc length of a curve is a fairly simple concept. We've already learned how to find the arc length of circles, but what about functions like cos(x+5) . . . Well first let's define arc length:

The length of a curve if were to be “rectified”.

"Rectified" is simply straightening out a curve from it's "curvy" shape to a linear line that can be measured. Now, you could do this with string, but you can also do it with calculus. Before we get into the calculus, here is a link to visualize a rectified curve: click here

So here is the calculus–again, nothing you don't already know how to do. Since, we're only going to go over how to find the arc length rectangularly

The first step is having a function that you would like to have the , and the formula.

For specifically rectangular form, ds is replaced with a formula showed below:

Since you already know how to find derivatives, the process is fairly simple from here on out. You can either use your graphing calculator by using "fnINT" of the rectangular formula above, or you can use Geogebra! Below is a 10-minute video I made to show how to take a real life object, and find the arc length of it. Not surprisingly, solving with calculus is much easier than measuring a curve with string.

If you would like to check out some examples of how to solve polar functions or parabolic functions, check out this website: click here. The formulas are also fairly simple, and easy to solve based on the calculus you know. Adios amigos.

Eliot