integral problem

Question

The velocity function is v(t)=t2−3t+2 for a particle moving along a line. Find the displacement and the distance traveled by the particle during the time interval [−3,6].

displacement = ??

distance traveled = ??

Answer 1

a) displacement is the integral from -3 to 6 of vdt or int from -3 to 6 of (t²-3t+2)dt = t³/3 - 3t²/2 +2t from -3 to 6

b) distance is the integral of f |v| dt. Easiest is to break up the integrals into intervals that are + or - v and add the absolute values of the integrals (i.e. if the velocity is negative in the interval, take that as a positive contribution to distance)

find the zeroes of the velocity: (t-1)(t-2) = 0 From -3 to 1, v>0 1 to 2, v<0 2+ v>0 So plug the endpoints into the integral expression in (a), but for 1 to 2, make the distance >0.