An “arc” is a curve segment; The arc length tells you how long this segment is.
Formal Definition of Arc Length
- 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) = x2/8 − ln(x) on the interval [1,2].
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) = x3/6 between ½ and 2.
Step 2: Insert the derivative into the formula for arc length:
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.
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 . 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 .
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 . 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 .
 Edwards, B. & Larson, R. (2009). Calculus, 9th edition. Cengage Learning.
 9.9 Arc Length. Retrieved April 12, 2021.
 Moll, V. et al. (2002). Bernoulli on arc length.
 Larson, R. & Edwards, B. (2009). Calculus, 9th Edition. Cengage Learning.
 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
Stephanie Glen. "Arc Length Formula" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/arc-length-formula/
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