Earlier this week, a teacher posted in the College Board’s AP Calculus Community a request for an explanation of computing the arclength of a curve without relying on formulas.

The following video is my proposed answer to that question. In it, I derive the fundamental arclength relationship before computing the length of from x=0 to x=3 four different ways:

As a function of x,

As a function of y,

Parametrically, and

As a polar function.

In summary, the length of any differentiable curve can be thought of as

where a and b are the bounds of the curve, the square root is just the local linearity application of the Pythagorean Theorem, and the integral sums the infinitesimal roots over the length of the curve.

To determine the length of any differentiable curve, factor out the form of the differential that matches the independent variable of the curve’s definition.