On Page 369, the textbook explains how the volume of a solid can be approximated by cylindrical solids. In each sentence, choose the correct option. V [ Select ] is [ Select] of the the exact volume solid. an approximation of the volume Ax [Select ] is the [ Select ] of base area the height k volume -th cylinder. A(xx) [Select ] is the (Select] of base area height the volume k -th cylinder. A(xx)Axk [Select ] base area is the (Select] of height volume the k -th cylinder. E A(x4)Ax¢ [ Select ] k=1 the exact volume an approximation of the volume is [ Select ] of the solid.
On Page 369, the textbook explains how the volume of a solid can be approximated by cylindrical solids. In each sentence, choose the correct option. V [ Select ] is [ Select] of the the exact volume solid. an approximation of the volume Ax [Select ] is the [ Select ] of base area the height k volume -th cylinder. A(xx) [Select ] is the (Select] of base area height the volume k -th cylinder. A(xx)Axk [Select ] base area is the (Select] of height volume the k -th cylinder. E A(x4)Ax¢ [ Select ] k=1 the exact volume an approximation of the volume is [ Select ] of the solid.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![6.1 Volumes Using Cross-Sections
369
cylindrical solid has a base whose area is A and its height is h, then the volume of the
cylindrical solid is
Volume = area × height = A•h.
In the method of slicing, the base will be the cross-section of S that has area A(x), and the
height will correspond to the width Axg of subintervals formed by partitioning the interval
[a, b] into finitely many subintervals [xx-1, X4 ].
Slicing by Parallel Planes
X-1
We partition [a, b] into subintervals of width (length) Axg and slice the solid, as we
would a loaf of bread, by planes perpendicular to the x-axis at the partition points
a = xo < x, <. .. < x, = b. These planes slice S into thin “slabs" (like thin slices of a
loaf of bread). A typical slab is shown in Figure 6.3. We approximate the slab between the
plane at x-1 and the plane at x by a cylindrical solid with base area A(xx) and height
Ax = x - x- 1 (Figure 6.4). The volume V of this cylindrical solid is A(x) · Ax4,
which is approximately the same volume as that of the slab:
FIGURE 6.3 A typical thin slab in the
solid S.
The approximating
cylinder based
on S(x) has height
Ax, - x - X-
Volume of the kth slab = V = A(x1) Axg.
Plane at x-1
The volume V of the entire solid S is therefore approximated by the sum of these cylindri-
cal volumes,
V
A(x1)
This is a Riemann sum for the function A(x) on [a, b]. The approximation given by this
Riemann sum converges to the definite integral of A(x) as n →o:
0|
Plane at x
lim
A(x)) Axz =
The cylinder's base
is the region S(x)
Therefore, we define this definite integral to be the volume of the solid S.
with area A(x,)
NOT TO SCALE
DEFINITION The volume of a solid of integrable cross-sectional area A(x)
from x = a to x = b is the integral of A from a to b,
FIGURE 6.4 The solid thin slab in
Figure 6.3 is shown enlarged here. It is
approximated by the cylindrical solid with
base S(x) having area A(x) and height
Ax
V =
A(x) dx.
X-1
This definition applies whenever A(x) is integrable, and in particular when A(x) is
continuous. To apply this definition to calculate the volume of a solid using cross-sections
perpendicular to the x-axis, take the following steps:
Calculating the Volume of a Solid
1. Sketch the solid and a typical cross-section.
2. Find a formula for A(x), the area of a typical cross-section.
3. Find the limits of integration.
4. Integrate A(x) to find the volume.
EXAMPLE 1
A pyramid 3 m high has a square base that is 3 m on a side. The cross-
section of the pyramid perpendicular to the altitude x m down from the vertex is a square
x m on a side. Find the volume of the pyramid.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5c95ede8-1154-42f9-b7dc-2ea41eb43732%2F2a85d8f9-357b-4dd5-a918-06b516b0cde4%2Fu480o8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:6.1 Volumes Using Cross-Sections
369
cylindrical solid has a base whose area is A and its height is h, then the volume of the
cylindrical solid is
Volume = area × height = A•h.
In the method of slicing, the base will be the cross-section of S that has area A(x), and the
height will correspond to the width Axg of subintervals formed by partitioning the interval
[a, b] into finitely many subintervals [xx-1, X4 ].
Slicing by Parallel Planes
X-1
We partition [a, b] into subintervals of width (length) Axg and slice the solid, as we
would a loaf of bread, by planes perpendicular to the x-axis at the partition points
a = xo < x, <. .. < x, = b. These planes slice S into thin “slabs" (like thin slices of a
loaf of bread). A typical slab is shown in Figure 6.3. We approximate the slab between the
plane at x-1 and the plane at x by a cylindrical solid with base area A(xx) and height
Ax = x - x- 1 (Figure 6.4). The volume V of this cylindrical solid is A(x) · Ax4,
which is approximately the same volume as that of the slab:
FIGURE 6.3 A typical thin slab in the
solid S.
The approximating
cylinder based
on S(x) has height
Ax, - x - X-
Volume of the kth slab = V = A(x1) Axg.
Plane at x-1
The volume V of the entire solid S is therefore approximated by the sum of these cylindri-
cal volumes,
V
A(x1)
This is a Riemann sum for the function A(x) on [a, b]. The approximation given by this
Riemann sum converges to the definite integral of A(x) as n →o:
0|
Plane at x
lim
A(x)) Axz =
The cylinder's base
is the region S(x)
Therefore, we define this definite integral to be the volume of the solid S.
with area A(x,)
NOT TO SCALE
DEFINITION The volume of a solid of integrable cross-sectional area A(x)
from x = a to x = b is the integral of A from a to b,
FIGURE 6.4 The solid thin slab in
Figure 6.3 is shown enlarged here. It is
approximated by the cylindrical solid with
base S(x) having area A(x) and height
Ax
V =
A(x) dx.
X-1
This definition applies whenever A(x) is integrable, and in particular when A(x) is
continuous. To apply this definition to calculate the volume of a solid using cross-sections
perpendicular to the x-axis, take the following steps:
Calculating the Volume of a Solid
1. Sketch the solid and a typical cross-section.
2. Find a formula for A(x), the area of a typical cross-section.
3. Find the limits of integration.
4. Integrate A(x) to find the volume.
EXAMPLE 1
A pyramid 3 m high has a square base that is 3 m on a side. The cross-
section of the pyramid perpendicular to the altitude x m down from the vertex is a square
x m on a side. Find the volume of the pyramid.
![On Page 369, the textbook explains how the
volume of a solid can be approximated by
cylindrical solids.
In each sentence, choose the correct option.
V
[ Select ]
is ( Select]
of the
the exact volume
solid.
an approximation of the volume
Axk
[ Select ]
is the
[ Select ]
of
base area
the
height
k
volume
-th cylinder.
A(xx)
[Select ]
is the ( Select ]
of
base area
height
the
volume
-th cylinder.
A(xx)Axk
[ Select ]
base area
is the (Select]
of
height
volume
the
k
-th cylinder.
> A(xx)Axk
[ Select ]
k=1
the exact volume
an approximation of the volume
is
[ Select ]
of the
solid.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5c95ede8-1154-42f9-b7dc-2ea41eb43732%2F2a85d8f9-357b-4dd5-a918-06b516b0cde4%2Flt7jtu_processed.jpeg&w=3840&q=75)
Transcribed Image Text:On Page 369, the textbook explains how the
volume of a solid can be approximated by
cylindrical solids.
In each sentence, choose the correct option.
V
[ Select ]
is ( Select]
of the
the exact volume
solid.
an approximation of the volume
Axk
[ Select ]
is the
[ Select ]
of
base area
the
height
k
volume
-th cylinder.
A(xx)
[Select ]
is the ( Select ]
of
base area
height
the
volume
-th cylinder.
A(xx)Axk
[ Select ]
base area
is the (Select]
of
height
volume
the
k
-th cylinder.
> A(xx)Axk
[ Select ]
k=1
the exact volume
an approximation of the volume
is
[ Select ]
of the
solid.
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