Parametric arc length is one of 2 parametric formulas in the calculus section of this library, and it is used at ap · university level.
Why parametric arc length works
In a tiny slice of time the point moves a little sideways and a little vertically, and those two moves are at right angles, so the actual travel is the hypotenuse. Pulling the time step out of that hypotenuse leaves the speed, and adding speed over time gives total distance covered.
What each symbol means
$t$ is the parameter and $[a,b]$ its interval.
Parametric arc length: when it holds
The derivatives should be continuous and the integral must converge.
When it stops applying
The integral measures distance travelled rather than the length of the picture drawn. If the parametrization backtracks or covers part of the curve twice, those stretches are counted again, so check that the parameter sweeps the curve exactly once.
Parametric arc length: a worked example
For $x=3t,y=4t$ on $[0,2]$, $L=\int_0^2 5dt=10$.
The mistake to avoid
What people do: Adding the horizontal and vertical rates instead of combining them under one root.
Why it goes wrong: Motion in two perpendicular directions combines by the Pythagorean theorem, so plain addition always overstates the distance travelled.
Do this instead: Square each rate, add the squares, and take a single square root of the total.
Parametric arc length: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $t$ is the parameter and $[a,b]$ its interval.
- Check the conditions before substituting. The derivatives 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
- Parametric
- 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 parametric arc length comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Derivative Rules — the lesson behind this formula: differentiate sums, products, quotients, and compositions.
- 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 parametric arc length
What if the curve is traced twice over the interval?
The integral returns double the length. Shorten the parameter interval so that each part of the curve is covered a single time.
Why is the integrand called the speed?
It is the length of the velocity vector built from the two rates, and speed multiplied by elapsed time is exactly distance travelled.
Does moving faster change the answer?
No, as long as the same curve is covered. A faster description runs over a shorter time interval, and the product of speed and time stays the same.
How do I extend this to a curve in space?
Add the squared rate of the third coordinate underneath the same square root. Nothing else about the setup changes.