Assume I have a path or trajectory over time
$$ \boldsymbol{s}(t) = [x_1(t), \dots, x_m(t)] $$
The velocity is
$$ \boldsymbol{v} = \frac{ds}{dt}\boldsymbol{e}_t = v \boldsymbol{e}_t$$ where $\boldsymbol{e}_t$ is a unit vector in the tangent direction. The magnitude of the velocity is
$$ v = \sqrt{\sum_{i = 1}^{m}{\left(\frac{dx_i}{dt}\right)^2}} $$
What is the interpretation or physical significance of the integral:
$$ \int{v dt} $$
If you integrate velocity as a function of time: $$ \int{v(t) dt} $$ you are left with your total distance traveled. Because velocity is your change in position as a function of time, integrating velocity as a function of time simply returns your total distance traveled. Similarly, the displacement can be found by evaluating $$ \int{|v(t)| dt} $$
As a general rule, if f(t) represents a change in something, then $\int_a^b{f(t) dt}$ will compute the total change from $t = a$ to $t = b$