4.1. DETERMINING DISTANCE TRAVELED FROM VELOCITY
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will be used in question (d).
(b) How far did the person travel during the two hours? How is this distance related
to the area of a certain region under the graph of y = v(t)?
(c) Find an algebraic formula, s(t), for the position of the person at time t, assuming
that s(0) = 0. Explain your thinking.
(d) On the right-hand axes provided in Figure 4.1, sketch a labeled graph of the
position function y = s(t).
(e) For what values of t is the position function s increasing? Explain why this is the
case using relevant information about the velocity function v.
⊲⊳
Area under the graph of the velocity function
In Preview Activity 4.1, we encountered a fundamental fact: when a moving object’s velocity
is constant (and positive), the area under the velocity curve over a given interval tells us
the distance the object traveled. As seen at left in Figure 4.2, if we consider an object
1
2
3
1
3
mph
hrs
v(t) = 2
A 1
1
2
3
1
3
mph
hrs
y = v(t)
A 2
Figure 4.2: At left, a constant velocity function; at right, a non-constant velocity function.
moving at 2 miles per hour over the time interval [1, 1.5], then the area A 1 of the shaded
region under y = v(t) on [1, 1.5] is
A 1 = 2
miles
hour
·
1
2
hours = 1 mile.
This principle holds in general simply due to the fact that distance equals rate times time,
provided the rate is constant. Thus, if v(t) is constant on the interval [a, b], then the