Let be a curve in . How would you find the length of the curve? One approach is to start by approximating the curve with segments.

Divide the base interval up into subintervals:

This is called a * partition* of the subinterval.
You may recall doing this to set up * Riemann
sums*.

If you plug the a's into f, you get points on the curve which you can connect with segments. Here's a pictures with 4 points and 3 segments:

The sum of the segments lengths approximates the length of the curve.

* Definition.* A curve is * rectifiable* if
the sums of the segment lengths have an * upper
bound*: that is, there is a number M such that for every
partitition of , the sum of the segment lengths
is less than M.

If a curve is rectifiable, the * length* of the
curve is least upper bound of the numbers M which bound the sums of
segment lengths for all partititions.

If a curve has reasonable properties, we can compute the length using an integral.

* Theorem.* Suppose is a curve where f is differentiable and
is continuous. Then f is rectifiable, and the length
of f is

While the proof is a little technical, we can see why this makes
sense. is the * speed* of an
object moving along the curve. In a small increment of time, the object moves a distance . If we let the time increments go to 0 and
add up the distances by integrating, we get the distance travelled by
the object, which is the length of the curve.

* Example.* Find the length of the curve

The length is

* Example.* Find the length of the curve

The length is

Copyright 2018 by Bruce Ikenaga