James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

406 Chapter 5 Integrals
directions, Fsbd 2 Fsad represents the net change in y.] So we can reformulate FTC2 in
words as follows.
Net Change Theorem The integral of a rate of change is the net change:
y b
F9sxd dx − Fsbd 2 Fsad
a
This principle can be applied to all of the rates of change in the natural and social sciences
that we discussed in Section 3.7. Here are a few instances of this idea:
● If Vstd is the volume of water in a reservoir at time t, then its derivative V9std is the
rate at which water flows into the reservoir at time t. So
y t2
V9std dt − Vst 2 d 2 Vst 1 d
t 1
is the change in the amount of water in the reservoir between time t 1 and time t 2 .
● If fCgstd is the concentration of the product of a chemical reaction at time t, then
the rate of reaction is the derivative dfCgydt. So
y t2
t 1
dfCg
dt
dt − fCgst 2 d 2 fCgst 1 d
is the change in the concentration of C from time t 1 to time t 2 .
● If the mass of a rod measured from the left end to a point x is msxd, then the linear
density is sxd − m9sxd. So
y b
sxd dx − msbd 2 msad
a
is the mass of the segment of the rod that lies between x − a and x − b.
● If the rate of growth of a population is dnydt, then
y t2
t 1
dn
dt dt − nst 2d 2 nst 1 d
is the net change in population during the time period from t 1 to t 2 . (The population
increases when births happen and decreases when deaths occur. The net
change takes into account both births and deaths.)
● If Csxd is the cost of producing x units of a commodity, then the marginal cost is
the derivative C9sxd. So
y x2
x 1
C9sxd dx − Csx 2 d 2 Csx 1 d
is the increase in cost when production is increased from x 1 units to x 2 units.
● If an object moves along a straight line with position function sstd, then its velocity
is vstd − s9std, so
2
y t2
vstd dt − sst 2 d 2 sst 1 d
t 1
is the net change of position, or displacement, of the particle during the time
period from t 1 to t 2 . In Section 5.1 we guessed that this was true for the case
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