It is really hard to help you when we dont know what f(t) stands for.
I am going to assume that f(t) is a function of position with respect to time.
Is the function of velocity at time t.
I guess that you put t=0, and got 36 feet/sec, but this is now what you are suppose
to do. is the function of velocity at time t.
Notice that when the particel is at rest (t=2) it is turning back, and stops again at t=6.
Then it turns around for the 2nd time, and goes back to s=32 (at t=8).
That means that the particle goes forward 32 feet then backwards 32 feet, and forward
again for 32 feet. So what is the total distance?
Here is plot of f(t)
Ask if there is something you don't understand.