In this post I want to explain how a definite integral can be used to find the arc length of a curve because I think that it is fascinating that integrals have functions other than just finding the area under curves. Let us start with a definition:

A **RECTIFIABLE **curve is one that has a finite arc length. A sufficient condition for the graph of a function to be rectifiable between and is that be continuous on . Such a function is **continuously differentiable** on , and its graph on the interval is a **smooth curve**.

Now let’s consider a function that is continuously differentiable on the interval . We can approximate the graph of by *n *line segments whose endpoints are determined by the partition:

By letting and , we can approximate the length of the graph by the following:

This approximation becomes more and more precise as . Therefore the length of the graph is

Because exists for each in , the Mean Value Theorem guarantees the existence of in such that

Because is continuous on , it follows that is also continuous and therefore integrable on , which implies that =