Now let's figure out how we could find the length of a curve.

Given a curve gamma, how do we find how long it is?

Normally, you would take maybe a piece of yarn,

lay it along the curve, then straighten it out and measure its length.

How do you actually do that?

Well, suppose we take this interval from a to b and

subdivide it again to its little pieces, and

look at this intermediate points on the curve, and we can approximate

the length of the curve by just measuring straight between all those points.

And the closer the points are together, the better the approximation seems to be.

So the length of gamma can be approximated by taking gamma of tj

plus 1 minus gamma of tj and the absolute value of that.

If the sum has a limit as n goes to infinity,

that is called the length of gamma and if this limit exists,

we say that the curve gamma is rectifiable or it has a length.

Note that not every curve has a length.

You could imagine that,

even though it seemed that this piece was a good approximation of this curve here.

As you zoom in really far, if you zoom into a little, little piece right here.

If you zoom into that, maybe there's a lot more going on than you actually thought

and it's a whole lot longer than you thought.

And then if you zoom into another little piece, that happens again.