Why is momentum the integral of force?

Newton's second law, $\vec{F} = m\vec{a}$, expressed in terms of momentum is $\vec{F} = \frac{d \vec{p}}{dt}$ (see text, pp. 224-225), i.e., force is rate of change of momentum. Thinking conceptually: force accelerates a massive object, and so changes its momentum (its ``importance of motion''). The calculus-inverse of ``force is derivative of momentum'' is `` momentum is integral of force'', i.e., $\Delta p = \int_i^f \vec{F} dt$.