Calculus How To

Arc Length Formula

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Contents:

  1. Arc Length
  2. Rectifiable Curve

What is an Arc Length?

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

arc length graph

Arc (yellow line) for the interval [½, 2].


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.
arc length formula

Where:

Arc Length Formula Example

Example Question: Find the arc length of f(x) = x2/8 − 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:
arc length example 1


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 basic function:

Example question: Find the arc length of f(x) = x3/6 between ½ and 2.

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

Step 2: Insert the derivative into the formula for arc length:
integral for arc length step 2

Simplifying:
simplifying the integral

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.

rectifiable curve

A cycloid, generated by a rolling curve, is a rectifiable curve. Image: Zorgit| Wikimedia Commons.


koch snowflake

Fractals, like this Koch snowflake, are nonrectifiable.


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 , via Wikimedia Commons.

CITE THIS AS:
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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