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 =