Arc length adds up infinitely many tiny straight pieces along a curve. Each piece is a hypotenuse.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The rectangular formula
The length is the integral of √(1 + (dy/dx)²) dx. The 1 comes from the horizontal step being dx.
The parametric formula
The length is the integral of √((dx/dt)² + (dy/dt)²) dt. Both steps now vary, so neither is 1.
Both are Pythagoras
Each formula is the hypotenuse of an infinitesimal right triangle. They differ only in what the steps are measured against.
Usually not evaluable
Arc length integrals rarely have elementary antiderivatives. The exam mostly asks you to set one up correctly.
Marks are for the setup
A correct integrand and correct limits earn the marks. Numerical evaluation is a calculator step.
Measuring the curve itself
Arc length is ∫√(1 + (dy/dx)²)dx. The square root comes from Pythagoras applied to a tiny piece of curve, whose length is √(dx² + dy²). The formula is Pythagoras integrated.
The parametric version
∫√((dx/dt)² + (dy/dt)²)dt is the integral of speed over time. It is the same formula with the parameter made explicit, and it handles curves that are not functions of x.
These integrals are usually intractable
The square root rarely yields to elementary techniques, so most arc length integrals are evaluated numerically. Setting up the integral correctly is what the exam asks for, more often than evaluating it.
Surface of revolution
Rotating a curve produces surface area 2π∫r·ds, where ds is the arc length element. The 2πr is the circumference traced by each point, and ds is the length it is traced along.
Step 2: Try It Yourself
Tap and try it out.
The hypotenuse is not given. It comes from 3² + 4² = 25, whose square root is 5.
Step 3: Watch an Example
One step at a time.
Watch Rosa Set Up an Arc Length
Rosa needs the length of y = x² from x = 0 to x = 2.
- Step 1
She differentiates: dy/dx = 2x.
Step 4: Your Turn
Practice makes it stick.
The Derivative
Problem 1 of 2
y = x². What is dy/dx at x = 3?
The Step
Problem 2 of 2
dx/dt = 3 and dy/dt = 4. What is √((dx/dt)² + (dy/dt)²)?
Set Up the Length
1 of 8
dx/dt = 6, dy/dt = 8. Speed?
2 of 8
dx/dt = 5, dy/dt = 12. Speed?
3 of 8
In the rectangular arc length formula, what constant sits under the root?
4 of 8
y = x², so dy/dx = 2x. What is dy/dx at x = 5?
5 of 8
y = 3x. What is dy/dx?
6 of 8
Do arc length integrals usually have elementary antiderivatives? 1 yes, 0 no.
7 of 8
Match each form with its integrand.
Tap a card on the left to start.
8 of 8
dx/dt = 8, dy/dt = 15. Speed?
Step 5: Quick Check
Show what you know.
Question 1 of 2
dx/dt = 9 and dy/dt = 12. What is the speed?
Question 2 of 2
Where does the arc length formula come from?
What You Learned
- Arc length integrates √(1 + (dy/dx)²) in rectangular form.
- In parametric form it integrates √((dx/dt)² + (dy/dt)²).
- Both are Pythagoras, and the marks are usually for the setup.