Segment of a Curve

A segment of a curve in the plane (Figure 6.3.1) is described by

y = f(x), a ≤ x ≤ b.

What is its length? As usual, we shall give a definition and then justify it. A curve y = f(x) is said to be smooth if its derivative f'(x) is continuous. Our definition will assign a length to a segment of a smooth curve.

Figure 6.3.1

Because

,

the equation is sometimes

written in the form

with the understanding that x is the independent variable. The length s is always greater than or equal to 0 because a < b and

JUSTIFICATION

Let s(u, w) be the intuitive length of the curve between t = u and t = w. The function s(u, w) has the Addition Property; the length of the curve from u to w equals the length from u to v plus the length from v to w. Figure 6.3.2 shows an infinitesimal piece of the curve from x to x + Δx. Its length is Δs = s(x, x + Δx).

Figure 6.3.2

The slope dy/dx is a continuous function of x, and therefore changes only by an infinitesimal amount between x and x + Δv. Thus the infinitesimal piece of the curve is almost a straight line, the hypotenuse of a right triangle with sides Δx and Δy. Hence

(compared to Δx). Dividing by Δx,

Then

(compared to Δx).

Using the Infinite Sum Theorem,

Example 1