Arc Lengths

Consider the length of each line segment:

So the total length of each segment is:

(see Summation and Product Notation)

We now have our arc length in the form of a Riemann Sum. Now we take limit on :

which are definite integrals

Definition

The arc length of a function is:

Example

A cable suspended between two supports of equal height assumes the form of a catenary:

Note that (from differentiation rules)

For , we use the definition of hyperbolic sine and cosine: