8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled

Arc length formula

For a smooth curve y = f(x) on [a, b], the arc length is L = ∫ from a to b of √(1 + (dy/dx)²) dx. This comes from summing tiny hypotenuses of right triangles with legs dx and dy, approximating the curve with infinitely many line segments.

Parametric arc length

If a curve is given parametrically by x(t) and y(t), the arc length over [a, b] is L = ∫ from a to b of √((dx/dt)² + (dy/dt)²) dt. This same formula gives distance traveled by a particle whose position is (x(t), y(t)).

Distance vs. displacement

For particle motion, distance traveled uses speed = √((dx/dt)² + (dy/dt)²) integrated over time, always giving a positive total. This differs from displacement, which is just the net change in position (final minus initial), and can be smaller or even zero.

Key terms

Arc length formula for y = f(x)

L = ∫ from a to b of √(1 + (f'(x))²) dx

Arc length formula for parametric curves

L = ∫ from a to b of √((x'(t))² + (y'(t))²) dt

Distance traveled by a particle

Integral of speed √((x'(t))² + (y'(t))²) over the time interval; always nonnegative.

Displacement vs. distance

Displacement is net position change (final − initial); distance traveled is the total path length, always ≥ displacement in magnitude.

Geometric origin of arc length formula

Approximating the curve with tiny line segments and applying the Pythagorean theorem: ds = √(dx² + dy²).

Try an AP-style question

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

1. ∫ₐᵇ √(1 + f(x)²) dx
2. ∫ₐᵇ √(1 + [f'(x)]²) dx
3. ∫ₐᵇ √(f'(x)) dx
4. ∫ₐᵇ (1 + f'(x)) dx