Arc length for a parametrically defined curve is computed by expressing the differential arc-length element ds = √(dx² + dy²) in terms of the parameter's differential, yielding ds = √((dx/dt)² + (dy/dt)²) dt, and then integrating this expression over the parameter's range rather than over x or y directly. This generalizes the rectangular-coordinate arc-length formula to curves defined by a parameter that is not itself one of the plotted variables, situating the concept within the calculus of parametric curves and connecting to the broader theory of computing geometric quantities (length, and by extension area/surface area) from parametrizations via the chain rule and differentials.
Arc length for a parametrically defined curve is computed by expressing the differential arc-length element ds = √(dx² + dy²) in terms of the parameter's differential, yielding ds = √((dx/dt)² + (dy/…