Speed

The magnitude of velocity, $|\mathbf{r}'(t)|$ — a positive number with no direction attached.

Speed is $|\mathbf{r}'(t)|=\sqrt{[f'(t)]^2+[g'(t)]^2+[h'(t)]^2}$ — the magnitude of velocity, not velocity itself: a positive scalar with no direction attached, exactly what a speedometer would read at that instant. Because it is never negative, integrating speed over time only ever accumulates, which is what makes it the integrand for arc length.

Examples

For $\mathbf{r}(t) = \langle 3\cos t, 3\sin t, 4t\rangle$ , $|\mathbf{r}'(t)| = \sqrt{9\sin^2t+9\cos^2t+16}=5$ — constant speed.