Arc Length of a Parametric Curve

The arc length can be computed without taking the square root of the sum of the squares of the derivatives of $x(t)$ and $y(t)$.

Both $dx/dt$ and $dy/dt$ are squared, summed, and then the square root of the result is integrated over the interval for $t$.

The arc length of the curve is given by $\int_{a}^{b} \sqrt{(dx/dt)^2 + (dy/dt)^2}\, dt$.

The limits of integration for computing the arc length correspond to the parameter values over which the curve is defined.