7
Themes for Advanced Placement Calculus
25
Theme
7
Volumes with Known Cross Sections and
Other Applications of Integration
Summary
Most of the applications of integration are based on the summation interpretation of definite integrals as
limits of Riemann Sums. This idea is highlighted on page 412 in the context of finding the area between two
curves
and
for
The disk method for finding volumes can be analyzed similarly (page 422). For a horizontal axis of rota
tion, the volume of a representative disk is
as indicated in the figure on the right.
The concept of work can also be thought of as a summation of increments (page 453)
Work
5
o
s
Force
ds
Distance
d
5
E
b
a
F
s
x
d
dx
.
Volume
5
o
s
Area of disk
ds
Thickness
d
5
E
b
a
p
f
R
s
x
dg
2
dx
R
s
x
d
2
D
x
,
Area
5
o
s
Height
ds
Width
d
5
E
b
a
f
f
s
x
d
2
g
s
x
dg
dx
a
≤
x
≤
b
.
g
s
x
d
,
f
s
x
d
Key to Text Coverage
Section
Examples
Exercises
Topics
4.3
1–6
1–71
Riemann sums, definite integrals and area
6.1
1–5
1–52, 57–81
Area between two curves
6.2
1–7
1–67
Volumes by disks and known cross sections
6.3
1–5
1–43
Volumes by the shell method
6.4
1–7
1–56
Arc length and surfaces of revolution
6.5
1–46
Work
x
ab
x
i
f
g
y
fx
(
)
i
(
)
i
gx
(
)
i
(
)
i
Height:
Width:
−
&
∆
&
x
∆
&
x
Representative rectangle
Rx
()
∆
&
x
a
b
V
=
π
&
∫
&
[()
]
2
dx
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 Spring '08
 Grether
 Calculus, Advanced Placement Calculus

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