A particle that moves along a straight line has velocity $\displaystyle v(t) = t^2e^{-2t}$ meters per second after t seconds. How many meters will it travel during the first t seconds?
Since the velocity in this case is always positive my following comment will not change the answer, but it should be mentioned.
To find distance you need to integrate the magnitude of the velocity. ie.
$\displaystyle \Delta s = \int dt \, |t^2 e^{-2t}|$
-Dan