# Applications of the Integral

Math
Summary

## Length of a Graph

Summary Length of a Graph

In addition to two-dimensional areas and three-dimensional volumes, the integral can be used to compute one-dimensional lengths. The idea, once again, is to approximate the length by a sum and to take the limit as the number of summands approaches to infinity.

More precisely, we want to calculate the length of the graph of a function f (x) from x = a to x = b. This length can be expressed as the sum of the lengths of the graph from x = a + (i - 1)Δx to x = a + iΔx, for i = 1,…, n, where Δx = (b - a)/n. We approximate the lengths of these smaller curves by line segments segments with the same endpoints, having lengths of Making a further approximation, we replace these segments with segments tangent to the graph at x = xi (with endpoints that have the same x-values as before), where xi is some number in the interval [a + (i - 1)Δx, a + iΔx]. The length of one of these new segments is equal to = Δx This is illustrated below.

 Δx    dx