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