
We can use the procedure of “Slice, Approximate, Integrate” to find the length of curves.

We have seen how the procedure of “Slice, Approximate, Integrate” can be used to find areas and volumes. Another geometric application of this procedure is to find the length of a segment of a curve.

Length of Curves Formula

Let’s see a few examples.

Just as it was sometimes advantageous to integrate with respect to $y$ in our area and volume calculations, it can also help us sometimes in arclength calculations. Unlike the area and volume problems, where the geometry of the region often suggested a preferred variable of integration, these problems require us only to consider how we describe the curve in question (and whether we want to work with its given description!) when choosing the variable of integration.

Sometimes, the integrals that arise can be tricky to compute analytically and require careful differentiation and algebra.

Finally, most of the integrands arising in length calculations do not have elementary antiderivatives, so oftentimes you will only be able to set them up and estimate them numerically.

Final thoughts

To summarize some important points from this section:

As usual, practice is important. Make sure you work through the problems slowly. A small mistake early on in the problem can produce catastrophic results.

“Did you hear about the mathematician who took up gardening? He only grows vegetables with square roots!” - Anonymous