**Contents:**

## What is an Arc Length?

An “arc” is a curve segment; The **arc length** tells you how long this segment is.

## Formal Definition of Arc Length

In calculus, the arc length is an approximated with straight line segments using a definite integral variation of the distance formula.

Where:

- s is the arc length,
- a, b are the integral bounds representing the closed interval [a, b],
- f′ is the first derivative.

## Arc Length Formula Example

**Example Question**: Find the arc length of f(x) = x^{2}/8 − ln(x) on the interval [1,2].

Step 1: **Find the first derivative** of the function. This solution uses the power rule and the derivative for natural log rule:

f′(x) = (x/4) – (1/x).

Step 2: **Insert the derivative into the arc length formula**. Don’t forget to add the integral bounds:

Step 3: **Evaluate the integral,** using the usual methods of integration or an online integral calculator (I used the one at integral-calculator.com).

Arc length = ln(2) + (3/8) ≈ 1.068.

## Notes on The Challenging Arc Length Formula

Unfortunately, many of the definite integrals required to calculate arc length are **extremely challenging or even impossible to compute **[1, 2]. You may want to use a calculator, like I did in the example above, to avoid the frustration of dealing with impossible-to-solve integrals, which happen a *lot* for the arc length formula.

For example, look at what happens to the following fairly simple function:

**Example question:** Find the arc length of f(x) = x^{3}/6 between ½ and 2.

Step 1: **Find the first derivative** of the function (this example uses the power rule):

Step 2: **Insert the derivative into the formula for arc length**:

Simplifying:

Step 3: **Evaluate the Integral.** At this point, the integral is impossible to evaluate using the “usual” methods of integration. Which means that we have to use other methods to approximate the arc length.

## Rectifiable Curve: Finite Arc Length

A **rectifiable curve** has a finite arc length.

**Nonrectifiable curves,** like fractals, have infinite length. They could also be called *curves that are rectifiable by straight lines*; their arc lengths can be expressed as elementary functions of their endpoints [3]. Many basic curves are rectifiable, including cycloids. Non-rectifiable curves include the Koch snowflake, circle and ellipse.

## Defining a Rectifiable Curve

Formally, we can define a rectifiable curve as follows:

If a first derivative for a function is a continuous function on the closed interval [a, b], the graph is rectifiable between (a, f(a)) and (b, f(b)). Functions that meet this criteria are continuously differentiable on the specified closed interval and have a smooth curve between the two points [4].

## Dealing With Nonrectifiable Curves

In calculus and real analysis, we’re mostly interested in rectifiable curves of finite length as they are straightforward to analyze. In a way, calculus can “force” a curve to be rectifiable: it’s length can be measured by very small line segments, leading to extremely accurate “approximations.”

The humble parabola is nonrectifiable, but we normally only deal with intervals of a parabola in calculus; In the rare case when you want to analyze a parabola over its entire (infinite) length, it can be made rectifiable by adding or subtracting its arc length to the length of an auxiliary parabola [3]. Nonrectifiable curves can also have their arc lengths expressed in terms of transcendental functions. Nonrectifiable curves also show as solutions to half-linear differential systems [5].

## References

[1] Edwards, B. & Larson, R. (2009). Calculus, 9th edition. Cengage Learning.

[2] 9.9 Arc Length. Retrieved April 12, 2021.

[3] Moll, V. et al. (2002). Bernoulli on arc length.

[4] Larson, R. & Edwards, B. (2009). Calculus, 9th Edition. Cengage Learning.

[5] Naito, Y. et al. (2018). Rectifiable and Nonrectifiable Solution Curves of Half-Linear Differential Systems. Math. Slovaca 68. No 3. 575-590. Retrieved April 11, 2021 from: https://search.proquest.com/openview/60993ad5055fb6bd783b70fcb571d77d/1?pq-origsite=gscholar&cbl=2038886

Cycloid image: Zorgit, CC BY-SA 3.0

**CITE THIS AS:**

**Stephanie Glen**. "Arc Length Formula" From

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