Greenwood press

40
INTEGRATION
The area between 
T = 20 cos

2π(x−14)
24
+ 70
and T = 80, which is the same as 

18
10
(20 cos

2π(x−14)
24
+ 70 − 80)dx

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entire solid is formed. For example, the volume, 
r, of a sphere can be represented 
by the equation 
v =
4
3
πr
3
, where 
r is the radius of the sphere. This equation can 
be determined by revolving a semicircle, 
y =
√
r
2
− x
2
, about the 
x-axis.
One really thin cross-sectional slice of the sphere can be represented by a
cylinder with radius 
y and thickness ∆x, as shown in the left-hand diagram
below. The volume of this cylindrical cross section, then, is 
v = πy
2
∆x. The
integral will accumulate the volume of all of these cylinders that stack up against
one another from 
x = –r to x = r.
Therefore the volume of a sphere can be represented by 
π

r
−r
(
√
r
2
− x
2
)
2
dx,
which simplifies to 
v =
4
3
πr
3
. This formula tells manufacturers how much metal 
is needed to create certain ball bearings. The formula is also useful for ice cream
store owners to determine how many cones they can serve with each container
of ice cream, assuming that they can convert cubic centimeter units to gallons.
Orange juice manufacturers can use this relationship to estimate the amount of
orange juice they will receive from a batch of fresh oranges. 
What about predicting the volume needed to juice other fruits that have non-
circular curves, such as lemons, apples, and pears? The process would be simi-
lar to calculating the volume of a sphere, except that an equation would need to
be developed to model the perimeter of the fruit. For example, if the core of a
pear is placed along the 
x-axis, a pencil can trace its perimeter in the first two
quadrants. A fourth-degree function can model the curvature of a pear, such as
y = –0.016x
4
− 0.094x
3
− 0.068x
2
− 0.242x + 3.132, and then rotated around