I am making my own example of the application of integral in physics for my task as follows:
the velocity of an object at the time $t$ is given by $v(t)=2t$, how far has the object traveled at $t=5$?
my solution is that I integrate the given velocity function $2t$ resulting in a distance function $s(t)=t^2$ from $t=0$ to $t=5$ which is equal to 25.
Is that correct?
If it is, what about we substitute $t$ with 5 in the function $v(t)=2t$ which makes $v=10$. To obtain $s$, we multiply $10.5=50$ .
perhaps, there is something wrong either with my example or my second thought, but what does make it different?