The arc length of a parametric curve x(t), y(t) from t=a to t=b is computed by which integral?

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

Answer: (A)

The arc length along a parametric curve is the integral of the speed along the path. For x(t) and y(t), the differential arc length is ds = sqrt((dx)^2 + (dy)^2). Since dx = x'(t) dt and dy = y'(t) dt, this becomes ds = sqrt((dx/dt)^2 + (dy/dt)^2) dt. Integrating from t = a to t = b sums all those tiny pieces, giving the arc length as ∫_a^b sqrt((dx/dt)^2 + (dy/dt)^2) dt. The other forms either omit one component or use a difference under the square root, which does not represent the true Euclidean length.