Question

The velocity v(t)= -t^2 + 6t - 8, find the displacement and distance traveled during the time interval [-1,5]?

Answer 1

For the total displacement during that interval, Integrate the velocity function vs time, for -1 < t < 5 s.

Note that v starts out negative at
t = -1, and then becomes 0 at t = 2 and 4 s. Those are "turnaround times". To get the distance travelled (in either direction), you have to add the absolute value of the displacements during these three intervals:
-1 to +2 s
2 to 4 s
4 to 5 s.
You will have to integrate v dt during each of those intervals and add the absolute values.