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, dS. At such a close range, dS is a straight line segment. It forms the hypotenuse of a right triangle with legs, dS and dy. The angle between dS and dx is dθ.
The ratio of dS to dx is sec dθ.
We can move dx 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 dx is dy.
We now have our general formula for the length of the segment, S.