Why is displacement ,instead of position, the integral of velocity?

Question

Why is the term displacement used as being the integral of velocity instead of position. The integral should cancel the derivative.

Answer 1

Consider the simple differential equation that relates the instantaneous position vector $s(t)$ to the velocity vector $v(t)$:

$v(t) = \frac{ds(t)}{dt}$

Integrating both sides and taking bounds gives:

$\int_{t_1}^{t_2}v(t)dt = \int_{t_1}^{t_2}\frac{ds(t)}{dt}dt$

The Fundamental Theorem of Calculus gives:

$\int_{t_1}^{t_2}v(t)dt = s(t_2) - s(t_1)$

A study of the right hand side should reveal that it is the difference between the position vectors at time $t_2$ ("final") and time $t_1$ ("initial"), and that is by definition the displacement the particle has undergone between those two time points. And this is why the displacement is related to the integral of velocity with respect to time.