Barrons AP Calculus - David Bock

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 + 3t^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) = 6t^2 − 18t + 12.

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

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

SOLUTIONS:

(a) Since v(t) = 6t^2 − 18t + 12 = 6(t − 1)(t − 2), we see that:

Thus, the total distance covered between t = 0 and t = 4 is

When we replace v(t) by 6t^2 − 18t + 12 in (2) and evaluate, we obtain 34 units for the total

distance covered between t = 0 and t = 4. This can also be verified on your calculator by

evaluating

This example is the same as Example 26, in which the required distance is computed by another

method.

(b) To find the displacement of the particle from t = 0 to t = 4, we use the FTC, evaluating