Arc Length

You’ll be able to

Compute the length of a curve using the arc-length integral
Set up arc length for a function y = f(x)
Recognize arc length as an accumulation of tiny hypotenuses

Adding up tiny straight pieces

Arc length measures how far you would travel along a curve from x = a to x = b. Zoom in on the curve and each tiny piece is nearly a straight segment — the hypotenuse of a right triangle with legs dx and dy. Its length is √(dx² + dy²) = √(1 + (dy/dx)²) dx. Summing all these hypotenuses gives the arc-length integral. The +1 under the root is the horizontal leg; the (dy/dx)² term is the vertical rise.

Arc length of y = f(x)

L = ∫ₐᵇ √( 1 + (dy/dx)² ) dx

Comes from the Pythagorean length √(dx² + dy²) of each infinitesimal piece, factored to pull out dx.

Setting it up (even when it will not integrate nicely)

To build the integral: differentiate to get dy/dx, square it, add 1, take the square root, and integrate over [a, b]. Arc-length integrands are notoriously messy — the square root rarely has an elementary antiderivative — so on the calculator-active portion of the AP exam you typically set up the integral and evaluate it numerically. Getting the setup exactly right, with correct dy/dx and bounds, is where the points are.

Worked example

Set up the integral for the arc length of y = x^(3/2) from x = 0 to x = 4.

1.Differentiate: dy/dx = (3/2)x^(1/2).
2.Square it: (dy/dx)² = (9/4)x.
3.Insert into the formula: L = ∫₀⁴ √(1 + (9/4)x) dx.
4.This one integrates: with u = 1 + (9/4)x, the result is (8/27)[(1 + (9/4)x)^(3/2)]₀⁴ = (8/27)(10^(3/2) − 1).

Answer: L = ∫₀⁴ √(1 + (9/4)x) dx = (8/27)(10^(3/2) − 1) ≈ 9.07. The setup is the key step; the evaluation may be numeric.

Tip

Remember it is 1 + (dy/dx)², not (dy/dx)² alone. The 1 represents the horizontal leg dx of each little triangle; dropping it collapses the hypotenuse to just the vertical rise and undercounts the length.

Checkpoint

Which integral gives the arc length of y = f(x) from x = a to x = b?

Checkpoint

For a curve with dy/dx = 2x, the arc-length integrand √(1 + (dy/dx)²) equals:

On the exam

Most arc-length integrals cannot be evaluated by hand, so on the calculator section, set up ∫√(1 + (dy/dx)²) dx precisely and let the calculator finish. The setup — correct derivative, correct bounds — is what the rubric rewards.

Answer the 2 checkpoints as you read.