JentooCalculus BC

AP® Calculus BC · Unit 8 · Topic 8.13

The Arc Length of a Smooth, Planar Curve

Arc length measures the true distance traveled along a curve, not just straight-line distance, by integrating √(1+(dy/dx)2) over the interval.

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A straight segment's length comes from one application of the Pythagorean theorem: distance = √((delta x)2+(delta y)2). A curve bends, so no single triangle captures it. The fix is to zoom in until the curve looks straight. Over a tiny stretch dx, the curve rises by roughly dy, and that tiny piece has length √(dx2+dy2). Factor out dx: √(1+(dy/dx)2) dx. Add up infinitely many of these slivers with an integral, and you get the arc length formula: L = integral from a to b of √(1+(f'(x))2) dx.

The square root is unavoidable because it comes straight from distance itself, which is never linear in slope. This is also why arc length is not just the integral of f(x): that integral measures signed area under the curve, a completely different quantity that ignores how steeply the curve climbs. A flat function and a wildly oscillating function can have the same area but very different lengths, because length depends on the derivative, not the height.

Arc length sits right after volumes by cross-section and disks or washers in this unit, and it uses the same slicing logic: cut the object into pieces small enough to treat as simple shapes, then integrate. It also sets up a problem worth noticing ahead of time: if the curve ever turns vertical, dy/dx blows up, and the integral in x fails. Rewriting the curve as x as a function of y, or using parametric arc length, rescues these cases.

Key terms

Arc length
The actual distance traveled along a curve between two points, measured by summing infinitesimal straight pieces.
Arc length integrand
The expression √(1+(dy/dx)2), built from the Pythagorean theorem on a tiny piece of curve.
Differential arc length
The infinitesimal length element ds = √(dx2+dy2), the building block of the arc length integral.
Parametric arc length
An alternate arc length formula using √((dx/dt)2+(dy/dt)2), useful when a curve turns vertical in x-y form.

Watch it explained

  1. How it worksHow does the Pythagorean theorem on one tiny piece of curve build the arc length integral?
  2. Worked exampleFind the arc length of y = x3/2 from x = 0 to x = 4.
  3. Worked exampleFind the length of the curve y = ln(cos(x)) from x = 0 to x = π/4.
  4. Common mistakeWhy isn't the arc length just the integral of the function itself?
  5. Going furtherWhat happens to the arc length integral when the curve turns vertical?

Check yourself

Your turn1 of 2
Find the arc length integrand (the function under the square root, before integrating) for y = 2x3/2, as a function of x.

End of topic.

Next: Distance Traveled Along a Curve →