Solids With Holes
The previous ideas and techniques can also be used to find the volumes of solids with holes in them.
If
A(x)
is the area of the face formed by a cut at
x
, then it is still true that the volume is
V =
⌡
⌠
a
b
A(
x
) dx
.
However, if
the solid has holes, then some of the faces will also have holes and a formula for
A(
x
)
may be more
complicated.
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5.1
Volumes
Contemporary
Calculus
9
Sometimes it is easier to work with two integrals and then subtract:
(i)
calculate the volume S
of the solid
without the hole,
(ii)
calculate the volume H
of the hole, and
(iii)
subtract
H
from S.
Example 10
: Calculate the volume of the solid in Fig. 26.
Solution:
The face for a slice at
x
, has area
A(
x
)
=
{
area of large circle
}
–
{
area of small circle
}
=
π
{
large radius
}
2
–
π
{
small radius
}
2
=
π
{
x
+ 1
}
2
–
π
{
1/
x
}
2
=
π
(
x
2
+ 2
x
+ 1 – 1/
x
2
).
Then
Volume =
⌡
⌠
a
b
A(
x
) dx
=
⌡
⌠
1
2
π
(
x
2
+ 2
x
+ 1 – 1/
x
2
)
dx
=
π
{
1
3
x
3
+
x
2
+
x
+ 1/
x
}

2
1
≈
18.33 .
Alternately, the volume of the solid with the large circular faces is
⌡
⌠
1
2
π
(
x
2
+ 2
x
+ 1
)
dx =
19
π
3
≈
19.90 ,
and the volume of the hole is
⌡
⌠
1
2
π
( 1/
x
2
) dx
=
π
2
≈
1.57
so the
volume we want is 19.90 – 1.57 = 18.33 .
Practice 7
:
Calculate the volume of the solid in Fig. 27 .
WRAP UP
At first, all of these volumes may seem overwhelming –– there are
so many possible solids and formulas and different cases.
If you
concentrate on the differences, it is very complicated.
Instead, focus
on the pattern of
cutting, finding areas of faces, volumes of slices,
and adding
.
With that pattern firmly in mind, you can reason your
way to the definite integral.
Try to make cuts so the resulting faces have regular shapes (rectangles,
triangles, circles) whose areas you can calculate.
Try not to let the complexity of the whole solid confuse
you.
Sketch the shape of
one
face and label its dimensions.
If you can find the area of
one
face in the
middle of the solid, you can usually find the pattern for all of the faces and then you can easily set up the
integral.
5.1
Volumes
Contemporary
Calculus
10
PROBLEMS
In problems 1 – 6, use the values given in the tables to calculate the volumes of the solids.
(Fig. 28 – 33)
Table 1:
box
base
height
thickness
Table 2:
box
base
height
thickness
(Fig. 28)
1
5
6
1
(Fig. 29)
1
5
6
2
2
4
4
2
2
5
4
1
3
3
3
1
3
3
3
1
4
2
2
1
Table 3:
disk
radius
thickness
Table 4:
disk
height
thickness
(Fig. 30)
1
4
0.5
(Fig. 31)
1
8
0.5
2
3
1
2
6
1
3
1
2
3
2
2
Table 5:
slice
face area
thickness
Table 6:
slice rock area min. area
thickness
(Fig. 32)
1
9
0.2
(Fig. 33)
1
4
1
0.6
2
6
0.2
2
12
1
0.6
3
2
0.2
3
20
4
0.6
4
10
3
0.6
5
8
2
0.6
3
Fig. 28
1
2
5.1
Volumes
Contemporary
Calculus
11
In problems 7 – 12, represent each volume as an integral and evaluate the integral.
7.
Fig. 34.
For
0
≤
x
≤
3, each face is a rectangle with
base 2 inches and height
5–
x
inches.
8.
Fig. 35.
For
0
≤
x
≤
3, each face is a rectangle
with base x inches and height
x
2
inches.