Arc length of a graph: what it means and how to use it
Add infinitesimal straight-line lengths along $y=f(x)$.
Arc length of a graph is one of 7 applications formulas in the calculus section of this library, and it is used at ap · university level.
Why arc length of a graph works
Across a tiny horizontal step the curve is nearly straight, and it climbs by the slope times that step. The little piece of curve is therefore the hypotenuse of a tiny right triangle, and factoring the horizontal step out of that hypotenuse is what produces the square root in the integrand.
What each symbol means
$f'$ is slope and $[a,b]$ is the horizontal interval.
Arc length of a graph: when it holds
$f'$ should be continuous and the integral must converge.
When it stops applying
The tiny-triangle picture needs a slope at every point, so a corner or a vertical tangent forces you to split the interval or treat the integral as improper. A curve that doubles back, such as a full circle, is not a single function of x and must be handled in pieces.
Arc length of a graph: a worked example
For $f(x)=0$ on $[0,5]$, $L=\int_0^5 1dx=5$.
The mistake to avoid
What people do: Putting the function itself under the square root instead of its derivative.
Why it goes wrong: Length depends on how steeply the curve rises, not on how high above the axis it happens to sit, so a graph shifted upward would wrongly change length.
Do this instead: Differentiate first, square the derivative, add 1, and only then integrate the root.
Arc length of a graph: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $f'$ is slope and $[a,b]$ is the horizontal interval.
- Check the conditions before substituting. $f'$ should be continuous and the integral must converge.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most calculus slips.
Where this formula fits
- Subject
- Calculus formulas — 42 entries in this library
- Topic
- Applications
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where arc length of a graph comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Fundamental Theorem of Calculus lesson — the lesson behind this formula: connect accumulated area with instantaneous change.
- Calculus Calculator — check your substitution and the value it produces.
- Study calculus — the subject guide that explains the ideas these formulas compress.
- Calculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 42 calculus formulas — the full grouped reference, or the complete formula library.
Questions about arc length of a graph
Why is there a 1 under the square root?
It is the horizontal part of each tiny step after squaring. A perfectly flat graph has zero slope, and the integrand collapses to 1, correctly returning the width of the interval.
Why do these integrals so rarely have exact answers?
Squaring a derivative and then taking a root usually builds an expression with no elementary antiderivative, which is why numerical methods are the normal approach.
What happens if I swap the two limits?
The integral comes out negative, since swapping limits reverses the sign. Length is a positive measurement, so keep the smaller limit on the bottom.
How does this relate to the parametric length formula?
Let the parameter be x itself. Then the horizontal rate is 1, and the parametric integrand simplifies to exactly this expression.