Arc Length of a Curve

Suppose that y=f(x), is a smooth curve on the interval [a,b]. Define and find a formula for the arc length L of the curve y=f(x) over the interval [a,b].

Arc Length y= f(x) x=a to x=b

The basic idea for defining arc length is to break up the curve into small segments, approximating the curve segments by line segments, add the lengths of the line segments to form a Riemann sum that approximates the arc length L, and take the limit of the Riemann sums to obtain an integral for L.

To obtain this idea, divide the interval [a,b] into n subintervals by inserting points x₁,x₂,x₃,...,x_n-1 between a and b. As shown in the figure above, let P₀, P₁, P₂,...,P_n be the points on the curve with x-coordinates a,x₁,x₂,...,x_n-1,b and join these points with straight line segments. These line segments form a polygonal path that we can regard as approximation to the curve y=f(x). As suggested by the Figure below, the Length L_k of the k^th line segment in the polygonal path is Length Lk of the kth line segment

Arc Length Polygonal Path

If we now add the lengths of these line segments, we obtain the following approximation to the length L of the curve

Length of all line segments

To put this in the form of a Riemann sum we will apply the Mean-Value Theorem. The theorem implies that there is a point any point in the kth subinterval between x_k-1 and x_k such that