Integrating velocity at a specific time to find position

[Lenjaku]

When I integrate v(t) after I calculated the t, what do I get?
Let;s say t=2.
do I get the position vector in this 2s?
If so it means the integral goes from 0 to t? or I still need to calculate from 2 to t?

I could simply integrate v from 0 to t then place 2 in the position vector but I got an ex here integrating after calculating the t...I knew I must not integrate vector after calculating the t since it won't be general anymore...but still there is a use for this it seems...

 

[Lenjaku]

I figure out I need to calculate 2 to t...

 

[haruspex]

$\int_{t=T_1}^{T_2}v(t).dt$ is the change in position from time T₁ to time T₂.