Arc Length of Function Graphs

Definition Let $f$ be continuous on $[a, b]$. For a partition $a = x_0 < x_1 < \dots < x_n = b$, let $P_i = (x_i, f(x_i))$. The length of the inscribed polygon is

The arc length of the graph is the least upper bound of $L(P)$ over all partitions, when this bound is finite, and the graph is then called rectifiable. Adding a point to a partition never decreases $L(P)$, by the triangle inequality. Because $f$ is continuous, the least upper bound also equals $\lim L(P)$ as the longest subinterval width tends to $0$. This is a theorem of analysis, and for the smooth functions of the next section it follows directly from the Riemann sum argument given there.

Definition A function $f$ is smooth on $[a, b]$ if $f'$ exists and is continuous on $[a, b]$, using one-sided derivatives at the endpoints. If $f$ is smooth on $[a, b]$, the length of its graph is

This follows from the mean value theorem. On each subinterval, $f(x_i) - f(x_{i-1}) = f'(x_i^*)\,\Delta x_i$ for some $x_i^*$ in $(x_{i-1}, x_i)$, so

This is a Riemann sum of a continuous function, so it tends to the integral as the longest subinterval width tends to $0$. If $f$ is continuous on $[a, b]$ and smooth on each of finitely many subintervals that cover $[a, b]$, then $f$ is piecewise smooth and its length is the sum of the lengths of the pieces.

Definition If the curve is described as $x = g(y)$ for $c \le y \le d$, with $g'$ continuous on $[c, d]$, its length is

Both formulas are cases of $ds = \sqrt{dx^2 + dy^2}$. Use whichever variable makes the derivative continuous and the integral simpler.

Definition Let $f$ be smooth on $[a, b]$. The arc length function measured from $a$ is

By the fundamental theorem of calculus, $s'(x) = \sqrt{1 + f'(x)^2} \ge 1$, so $s$ is strictly increasing, and in differential form $ds^2 = dx^2 + dy^2$.

Definition For many smooth functions $f$, even simple ones, the integrand $\sqrt{1 + f'(x)^2}$ has no elementary antiderivative. The arc length still exists as a definite integral of a continuous function, and it is found by numerical integration, such as Simpson’s rule, to any desired accuracy.