Barrons AP Calculus

(a)

(b)

is equivalent to dt; similarly, if v(t) 0 on [a, b] (P moves only in the

negative direction), then (1) yields If v(t) changes sign on [a, b] (i.e.,

the direction of motion changes), then (1) gives the total distance traveled.
Suppose, for example, that the situation is as follows:

a t c v(t)   0;

c t d v(t)      0;

d t b v(t)   0.

Then the total distance traveled during the time interval from t = a to t = b is
exactly

Displacement

The displacement or net change in the particle’s position from t = a to t = b is
equal, by the Fundamental Theorem of Calculus (FTC), to

Example 1 __

If a body moves along a straight line with velocity v = t 3 + 3 t 2 , find the distance
traveled between t = 1 and t = 4.

SOLUTION:

Note that v > 0 for all t on [1, 4].

Example 2 __

A particle moves along the x-axis so that its velocity at time t is given by v(t) =
6 t 2 − 18t + 12.

Find    the total   distance    covered between t   =   0   and t   =   4.

Find    the displacement    of  the particle    from    t   =   0   to  t   =   4.