Calculus How To

Arc Length Formula

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Calculus Definitions >


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 simple 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.

References

[1] Edwards, B. & Larson, R. (2009). Calculus, 9th edition. Cengage Learning.
[2] 9.9 Arc Length. Retrieved April 12, 2021.

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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