Arc Length (Cartesian)

Definition

The arc length $L$ of a smooth curve $y = f(x)$ from $x = a$ to $x = b$ is defined as: $L = \int_{a}^{b} \sqrt{1 + [f'(x)]^2} \, dx$ A curve is smooth if $f'$ is continuous on the interval $[a, b]$.

• How to read: “L equals the integral from a to b of the square root of the quantity one plus f prime of x squared, with respect to x.”
• Meaning: Sum of infinitesimal hypotenuses along the curve—Pythagorean theorem applied to tiny steps $dx$ and $dy = f'(x)\,dx$.

Why It Matters

It allows for the precise measurement of distances along non-linear paths, which is critical for engineering structures and mapping terrain. Without it, we would be limited to measuring straight lines in a curved world.

Core Concepts

• Smoothness: For the formula to be valid, the derivative must be continuous, ensuring the curve has no sharp corners.
• Arc Length Differential: The infinitesimal length $ds$ is $\sqrt{dx^2 + dy^2}$, which simplifies to $\sqrt{1 + (dy/dx)^2} \, dx$.
  • How to read: “ds equals the square root of dx squared plus dy squared; or the square root of the quantity one plus the square of dy dx, all times dx.”
  • Meaning: $ds$ is the length of one microscopic segment; integrating gives total path length.
• Pythagorean Basis: The formula is derived from the limit of a sum of many short straight-line segments (hypotenuses) along the curve.

Connected Concepts

• Arc Length in Space
• Surface Area of Revolution (Cartesian)