What do we mean by the length of a curve? We might think of fitting a piece of string to the curve and then measuring the string against a ruler. But that might be difficult to do with much accuracy if we have a complicated curve. We need a precise definition for the length of an arc of a curve, in the same spirit as the definitions we developed for the concepts of area and volume.
If a curve is a polygon, we can easily find its length; we just add the lengths of the line segments that form the polygon. We are going to define the length of a general curve by first approximating it by a polygonal path (a path consisting of connected line segments) and then taking a limit as the number of segments of the path is increased. This process is familiar for the case of a circle, where the circumference is the limit of lengths of inscribed polygons.
Now suppose that a curve C is defined by the equation $y = f(x)$, where $f$ is continuous and $a ≤ x ≤ b$. We obtain a polygonal approximation to $C$ by dividing the interval $[a, b]$ into $n$ subintervals with endpoints $x_0, x_1, \ldots, x_n$ and equal width $Δx$. If $y_i = f(x_i)$, then the point $P_i(x_i, y_i)$ lies on $C$ and the polygonal path with vertices $P_0, P_1, \ldots, P_n$, is an approximation to $C$.
The length $L$ of $C$ is approximately the length of this polygonal path and the approximation gets better as we let $n$ increase. Therefore we define the length $L$ of the curve $C$ with equation $y = f(x), a ≤ x ≤ b$, as the limit of the lengths of these approximating polygonal paths (if the limit exists):
$$ L = \lim_{n \to \infty} \sum_{i=1}^{n} |P_{i-1}P_i| $$
where $|P_{i-1}P_i|$ is the distance between the points $P_{i-1}$ and $P_i$.
Notice that the procedure for defining arc length is very similar to the procedure we used for defining area and volume: We divided the curve into a large number of small parts. We then found the approximate lengths of the small parts and added them. Finally, we took the limit as $n → ∞$.
The definition of arc length is not very convenient for computational purposes, but we can derive an integral formula for L in the case where f has a continuous derivative.
If we let $\Delta y_i = y_i - y_{i-1}$, then
$$ |P_{i-1}P_i| = \sqrt{(x_i-x_{i-1})^2 + (y_i-y_{i-1})^2} = \sqrt{(\Delta x)^2 + (\Delta y_i)^2} $$
By applying the Mean Value Theorem to f on the interval $[x_{i-1}, x_i]$, we find that there is a number $x_i^*$ between $x_{i-1}$ and $x_i$ such that
$$ f(x_i) - f(x_{i-1}) = f'(x_i^*)(x_i - x_{i-1}) $$
that is,
$$ \Delta y_i = f'(x_i^*),\Delta x $$
Thus we have
$$ |P_{i-1}P_i| = \sqrt{(\Delta x)^2 + (\Delta y_i)^2} = \sqrt{(\Delta x)^2 + [f'(x_i^*),\Delta x]^2} $$
$$ = \sqrt{1+[f'(x_i^)]^2}\sqrt{(\Delta x)^2} = \sqrt{1+[f'(x_i^)]^2},\Delta x \qquad \text{(since } \Delta x > 0\text{)} $$
Therefore,