An object moves with velocity v(t)=t^2-8t+7
a) write a polynomial expression for the position of the particle at any time t greater or equal to zero.

b) at what time(s) is the particle changing direction

c) find the total distance traveled by the particle from t=0 and t=4

a) write a polynomial expression for the position of the particle at any time t greater or equal to zero. 
Position is found by integrating velocity: 

s(t) = (t^3)/3 - 4t^2 + 7t + c 
where c is a constant corresponding to the position at t=0. 

b) at what time(s) is the particle changing direction
the particle changes direction whenever the velocity is zero; the velocity function equals 

(t-1)(t-7) a difference of squares so the zeros are 1 and 7, it changes direction at 1 second and 7 seconds. 

c) find the total distance traveled by the particle from t=0 and t=4
s(0) = c
s(1) = 8/3 + c
s(4) = 64/3 - 64 + 28 + c. 

from 0 to 1 the particle travels 8/3 units. From 1 to 4 it travels -(64/3 - 36 - 8/3) = (-(56/3 - 108/3)) 
=-(-52/3) = 52/3 units 

so in total it travels 52/3 + 8/3 =20 units