The distance traveled by a particle along a parametric path r(t) from t = a to t = b is given by which integral?

• A. ∫_a^b sqrt((dx/dt)^2 + (dy/dt)^2) dt
• B. ∫_a^b (dx/dt + dy/dt) dt
• C. ∫_a^b (dx/dt)^2 + (dy/dt)^2 dt
• D. ∫_a^b sqrt((dx/dt)^2 - (dy/dt)^2) dt

Answer: (A)

Distance traveled along a parametric path comes from integrating the speed over time. If the path is r(t) = ⟨x(t), y(t)⟩, then the velocity is r′(t) = ⟨dx/dt, dy/dt⟩ and the speed is its magnitude |r′(t)| = sqrt((dx/dt)² + (dy/dt)²). So the distance from t = a to t = b is ∫_a^b |r′(t)| dt = ∫_a^b sqrt((dx/dt)² + (dy/dt)²) dt.

The other expressions don’t measure length: summing the rates, dx/dt + dy/dt, would give Δx + Δy, not the distance traveled. Integrating the sum of squares isn’t a length either, and the form with a minus sign under the square root isn’t generally valid for a path’s length.