# The Length of a Curve

## The arc length formula using infinitesimals

We wish to derive a general formula for the arc length of a curve given by the function, *y=f(x)*. We will do so using infinitesimals. These are infinitely small portions of the curve.

The figure below shows such a curve. The length of the segment between *f(x₀)* and *f(x₁)* is *S*. The circle to the right shows an infinitely small portion of the segment, d*S*. At such a close range, d*S *is a straight line segment. It forms the hypotenuse of a right triangle with legs, d*S* and d*y*. The angle between d*S* and* *d*x* is d*θ*.

The ratio of d*S* to* *d*x *is sec* *d*θ.*

We can move d*x *to the right-hand side of the expression. Once we integrate between *x₀* and *x₁, *we have an expression for *S.*

We have a problem. The variable of integration is *θ*, whereas the differential is in terms of *x*. This won’t do. We begin by translating the integrand into a more useful form.

This form is more useful to us. We can take tan² *θ *and express it in terms of d*x* is d*y.*

We now have our general formula for the length of the segment, *S*.