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INTEGRATION
39
A comparison of the areas between
the graph of 
f (t) = 6.37e
−0.04t
and the t axis on the interval from 
t = 0 to 2 hours (the light shaded
region) and from t = 6 to 8 hours
(the dark solid region).

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Integration can be used to help solve differential equations in order to for-
mulate new equations that compare two variables. A differential equation is a
relationship that describes a pattern for a rate. For example, the differential equa-
tion describing the rate of the growth of a rabbit population is proportional to the 
amount present and would be represented by the equation 
dP
dt
= kP , where P is
the population, 
t is the amount of time, and k is a constant of proportionality. If
there were 200 rabbits in the population seven months ago, and 500 rabbits in the
population right now, then an integral will help you find an equation that relates
the population of rabbits to the amount of time that has passed. In this case, solv-
ing the differential equation will result in a general equation of 
P = 200e
0.131t
,
where
t is the number of months that have passed since the rabbits were origi-
nally counted. This information can help farmers understand how their crops will
be affected over time and take preventative measures, since they will be able to
predict future rabbit populations, assuming that changes will not result in the
growth rate due to disease or removal.
The equation 
d = 0.5gt
2
+ v
o
t + d
o
is commonly used in physics when
studying kinematics to describe the vertical position, 
d, of an object based on the
time the object has been in motion, 
t. Values that are commonly substituted into
this equation are 
g = –9.8 meters per second squared to represent the accelera-
tion due to earth’s gravity, the initial velocity of the object, 
v
o
, and the initial ver-
tical position of the object, 
d
o
. How was this equation determined? Integration
can help explain how this expression is derived.
The acceleration of an object in vertical motion is equal to the constant value,
g, neglecting any air resistance. Acceleration is a rate of velocity, v, so v =
 gdt.
The velocity at 
t = 0 is v
o
, so this information and the integral determines the 
equation
v = gt + v
o
. Velocity is a rate of position, so 
d =
 (gt + v
o
)dt. The
vertical position at 
t = 0 is d
o
, so this information and the integral determine the
equation
d = 0.5gt
2
+ v
o
t + d
o
.
Many volume formulas in geometry can also be proven by integration. In this
case, the integral serves as an accumulator of small pieces of volume until the