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INTEGRATION
41
A sphere with radius, r, is formed
when rotating the semicircle
y =
√
r
2
− x
2
about the x-axis.
cylindrical slice inscribed in a sphere
cylindrical slices stacked to form a sphere
Cylindrical slices with radius 
y =
√
r
2
− x
2
and height 
∆x are stacked together 
to form the volume of the sphere.

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the
x-axis to form the solid, as shown below. An integral set up like the volume
of a sphere, and bounded by the 
x-intercepts of the function, will approximate
the volume of the pear. 
This pear has a volume 
≈ 352 milliliters, as a result of evaluating 
π

2.502
−6.347
(–0.016x
4
− 0.094x
3
− 0.068x
2
− 0.242x + 3.132)
2
dx.
online sources for further exploration
BHS calculus student projects

The case of the murky mell

CO
2
concentrations in a river

Flood levels

Gavin’s calculus projects

Heating-degree-days

Internet differential equations activities: Current projects

Modeling population growth

Nuclear medicine

Petroleum collection

Surveying

Tunnel Vision, Inc.

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42
INTEGRATION
A pear can be constructed by rotating the
function
y = –0.016x
4
− 0.094x
3
− 0.068x
2
−0.242x + 3.132 about the x-axis.

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INVERSE
(
MULTIPLICATIVE
)
43
INVERSE
(
MULTIPLICATIVE
)
A relationship in the form 
y =
k
x
, where 
k is a constant, is called an inverse
function. Sometimes you will see this relationship written as “
y is inversely pro-
portional to 
x.” The graph of this function is a hyperbola, but most real-world
applications with inverse functions relate only to nonnegative values in the
domain.
An inverse proportion indicates that the dependent variable decreases as the
independent variable increases, or vice versa. In other words, as one factor
changes, the other factor will change in the reverse direction. For example, pres-
sure is inversely proportional to the volume of an object. When you dive under-
water, the amount of air space in between your ears begins to decrease, causing
them to pop, because the pressure gradually increases. If your ears do not pop
and release the air inside, then you will feel discomfort or pain, because the pres-
sure becomes too great.
Bottling companies use this same principle in packaging their soft drinks. Air
and carbon dioxide are compressed in a small volume of space when you first
open a container. The built-up pressure inside the small amount of space will
cause the container to fizz or make a sound when it is first opened. After the
gasses in the container have been released and part of the bottle is emptied, the
pressure on the bottle decreases, since the air volume inside has increased. Thus
the bottle does not fizz as much when it is opened later on.
Size is influenced by a combination of surface area and volume. The ratio of
surface area to volume is an inverse relationship, because area units are squared
and volume units are cubed. The ratio of squared units to cubic units is equal to
inverse units. For example, the surface area of a cube with an edge length equal
to 2 cm is 24 cm
2
(6 square faces, each with area of 4 cm
2
). The volume of this 
cube is 8 cm
3
. The surface-area-to-volume ratio is 3 cm
–1
(determined by 
24 cm
2
8 cm
3
).
Notice that the units are a multiplicative inverse, or reciprocal, of cm. 
Since an organism’s metabolic rate is affected by this ratio, it can be mod-
eled after an inverse proportion. This means that large animals will typically
have slower metabolisms than smaller animals, because the ratio of surface area
to volume will decrease for larger volumes. Conversely, smaller animals will
have higher metabolic rates than larger animals, because this ratio increases for
A graph of the inverse function 
y =
4
x
.

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smaller volumes. Therefore rodents and dogs are much more likely to lose heat
from their bodies than bears and elephants, because they have less heat stored
inside. As a result, smaller animals need to be more active to maintain appro-
priate heat levels within their bodies, causing their metabolism to remain at high
levels. Animals and plants have naturally developed parts of their body to help
expand their surface area without adding considerable volume so that they can
increase their metabolic rate. For example, trees develop leaves from branches,
and humans use capillaries to extend their circulatory system. Microvilli, the
lining of the small intestine, is an example of a large surface area in the human
body with little volume, because it stretches to lengths of over seven meters
long!
Fuel consumption as a function of gas mileage is an inverse relationship. As
automobiles increase their fuel efficiency, or the number of miles per gallon they
attain while driving, then the gasoline consumers will purchase less fuel. Smaller
compact cars typically obtain better gas mileage, because there is less mass to
move when compared to less-fuel-efficient cars such as vans, trucks, and sport-
utility vehicles. If Americans drive approximately 10
12
miles each year, then the
fuel consumption of the United States each year can be represented by the func-
tion
g =
10
12
m
, where 
m is the average gas mileage of the cars that year. 
Production rates also form inverse relationships. The time it takes to com-
plete a task is inversely proportional to the rate at which an item is produced or
performed. For example, a grocery store clerk needs to staple price stickers on
5,000 cans. The amount of time needed for the job, 
t, is dependent on his pro-
ductivity rate, 
r, according to the function t =
5000
r
. If he works at a faster rate, 
then the job will take less time to complete.
Some people claim that “Murphy’s law” can be described as an inversely
proportional relationship. This law maintains that anything can go wrong at the
worst possible moment. For example, when constructing a stage, Murphy’s law
might strike if the most vital tool to complete the job is missing. Another exam-
ple is staying up all night to complete a term paper, only to realize that your disk
has gone bad or your printer is out of ink. If this law were described as an inverse
function, then the availability of an item or luck is inversely proportional to its
importance. Thus as an event or object becomes more important, Murphy’s law
can strike, indicating that it will likely not occur or be available. Conversely, the
object or phenomenon will more likely occur or become available when it is less
needed.
online sources for further exploration
Best practices in network security

Boyle’s law

Ears, altitude and airplane travel