Calculus Of One Real Variable – By Pheng Kim Ving
Chapter 12: Applications Of The Integral – Section 12.6: Arc Length

12.6
Arc Length

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1. Arc Length

Let C be a curve joining points A and B. See Fig. 1.1. Choose the points P₀, P₁, P₂, ..., P_n_–1, P_n along C such that A =
P₀ and B = P_n. Denote the length of the chord (line segment) P₀P₁ by |P₀P₁|, that of the chord P₁P₂ by |P₁P₂|, etc, ie,

Fig. 1.1

Length of curve or arc C is limit of sum of lengths of chords
as length of every chord approaches 0.

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2. Lengths Of Graphs Of Functions

The derivation of the formula for the length of the graph of a function f requires that f is differentiable and f ' is
continuous, as seen below. A function f is said to be continuously differentiable if it's differentiable on its domain and
its derivative f ' is continuous there. Let f be a continuously differentiable function and [a, b] a sub-interval of dom( f ).
See Fig. 2.1. We wish to find the length of the portion of the graph of f over [a, b].

Let s be the length of the portion of the graph of f over [a, b]. Form a regular partition of order n of [a, b]. We select
n = 5 as an example in Fig. 2.1. The partition points are a = x₀ < x₁ < x₂ < ... < x_n_–1 < x_n = b. So the sub-intervals [x_i_–1,

Fig. 2.1

Length s of graph of f over [a, b] is:

The length s of the graph of continuously differentiable f over [a, b] is:

Let y = f(x). Since f '(x) = dy/dx, Eq. [2.1] is equivalent to:

Remark 2.1

If 2 functions differ by a constant on [a, b], as seen in Fig. 2.2, then clearly the portions of their graphs there have an
equal length. This fact is confirmed by Eq. [2.1]. As the functions differ by a constant on [a, b], they have the same
derivative there, so, by Eq. [2.1], the portions of their graphs there have an equal length.

Fig. 2.2

If functions differ by a constant on [a, b] then portions of
graphs there have an equal length.

Example 2.1

Find the length of the graph of f(x) = x^2/3 over [1, 8]. Give the answer in approximate decimal format.

Solution

EOS

Note that we don't have to sketch the arc or curve in question. When finding areas and volumes, the sketch is needed to
correctly set up the integral. Here, it's not. However, it may be necessary to determine the characteristics of the shape of
the curve; see Problem & Solution 5.

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3. Differential Of Arc Lengths

The graph of f and the regular n-order partition of [a, b] in Fig. 2.1 are re-produced in Fig. 3.1. Consider the Riemann

Fig. 3.1 as an example for [x_i_–1, x_i]. Recall from Section 4.3 Definitions 2.1 that, at x_i_–1, the differential of x equals the

Fig. 3.1

Differential of arc length for f is:

Remark that here we're dealing with segments of tangent lines at x_i_–1, while in Part 2 we're dealing with chords P_i_–1P_i..
The arc length element is the length of a segment of a tangent line, not the length of a chord. As the arc length element
is obtained by operations on differentials (of x and y), it's also called the differential of arc length.

Definition 3.1

Suppose f is a continuously differentiable function and let y = f(x). Then the differential ds of arc length for f is:

We've seen above that:

The arc length is the definite integral of the differential of arc length.

The small right triangle on the graph of f in Fig. 3.1 has all its 3 sides expressed in differential terms, and thus is called a
differential triangle. We can manipulate Eq. [3.1] as follows:

The differential ds of arc length for continuously differential f where y = f(x) is also given by:

From Eqs. [3.2] and [3.3] we obtain:

But this is exactly Eq. [2.2]. We observe that the differential triangle with its equation:

serves as a useful mnemonic device to obtain Eqs. [2.1], [2.2], or [3.4].

Example 3.1

Calculate the length of the graph of:

from x = 1 to x = 2, using differential of arc length in forming the integral.

Solution
The differential of arc length is:

The required arc length is:

EOS

When asked to use the differential of arc length in forming the integral, we use Eqs. [3.3] and [3.4]. For the function in
this example, we factor 1 + (dy/dx)² into the square of something, and so the radical (square root symbol) in the
expression for ds is removed.

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4. For Functions Of y

Fig. 4.1

Length of graph of continuously differentiable function x = g( y) over [c, d] is:

We can do this reversing because g is a continuously differentiable function of the independent variable y and the
dependent variable is x where x = g( y), and because the formulas for arc length discussed above are valid for any
continuously differentiable function, regardless of the variable. Eq. [3.1] can also be manipulated as follows:

So:

Example 4.1

Compute the length of the graph of:

from y = 1 to y = 2.

Solution

The differential of arc length is:

The desired arc length is:

EOS

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5. Avoiding Confusion

To avoid the possible confusion as to which differential dx or dy to use in the equation for arc length, we make the
following observation. For arc length calculated along the x-axis from x = a to x = b, the function is a function of the
variable x, so it's the derivative with respect to x and the differential of x, ie ( f '(x) or dy/dx) and dx. For arc length
calculated along the y-axis from y = c to y = d, the function is a function of the variable y, so it's the derivative with
respect to y and the differential of y, ie ( g '( y) or dx/dy) and dy.