5128_Ch07_pp378-433

380 Chapter 7 Applications of Definite Integrals
Reminder from Section 3.4
A change in position is a displacement.
If s(t) is a body’s position at time t, the
displacement over the time interval
from t to t Δt is s(t Δt) s(t). The
displacement may be positive, negative,
or zero, depending on the motion.
constant during a motion, we can find the displacement (change in position) with the
formula
Displacement rate of change time.
But in our case the velocity varies, so we resort instead to partitioning the time interval
0, 1 into subintervals of length Δt so short that the velocity is effectively constant on
each subinterval. If t k is any time in the kth subinterval, the particle’s velocity throughout
that interval will be close to vt k . The change in the particle’s position during the brief
time this constant velocity applies is
vt k Δt. rate of change time
If vt k is negative, the displacement is negative and the particle will move left. If vt k
is positive, the particle will move right. The sum
vt k Δt
of all these small position changes approximates the displacement for the time interval
0, 1.
The sum vt k Δt is a Riemann sum for the continuous function vt over 0, 1. As
the norms of the partitions go to zero, the approximations improve and the sums converge
to the integral of v over 0, 1, giving
Displacement 1
vt dt
0
( t2
0
1
) dt
t
8
1 2
]
[ t 3
8
3 t 1
1 3 8 2 8 1 1
.
3
1
0
During the first second of motion, the particle moves 113 cm to the left. It starts at
s0 9, so its position at t 1 is
New position initial position displacement 9 1 1
1 6 .
3 3
(b) If we model the displacement from t 0 to t 5 in the same way, we arrive at
Displacement 5
5
vt dt
[ t 3
8
3
0
t
]
35.
1
0
The motion has the net effect of displacing the particle 35 cm to the right of its starting
point. The particle’s final position is
Final position initial position displacement
s0 35 9 35 44.
]
Support Graphically The position of the particle at any time t is given by
st t
8
[ u2
0 u 1 2 du 9,
because st vt and s0 9. Figure 7.2 shows the graph of st given by the
parametrization
xt NINT vu, u, 0,t 9, yt t, 0 t 5.
continued