Volume: The Disc Method

1
Volume The Disc Method

• Section 6.2

2

• If a region in the plane is revolved about a
  line, the resulting solid is a solid of
  revolution, and the line is called the axis of
  revolution. The simplest such solid is a right
  circular cylinder or disc, which is formed by
  revolving a rectangle about an axis .

3
Revolving a Function

• Consider a function f(x) on the interval a, b
• Now consider revolvingthat segment of curve
  about the x axis
• What kind of functions generated these solids of
  revolution?

f(x)
a
b
4
Disks
f(x)

• We seek ways of usingintegrals to determine
  thevolume of these solids
• Consider a disk which is a slice of the solid
• What is the radius
• What is the thickness
• What then, is its volume?

5
If a region of a plane is revolved about a line,
the resulting solid is the solid of revolution,
and the line is the axis of revolution. The
simplest is a right circular cylinder or disk, a
rectangle revolved around the x-axis. The
volume is equal to the area of the disk times
the width of the disk, V pr2w.
r
w
6
The Disc Method

• Volume of disc
• (area of disc)(width of disc)

7
Disks

• To find the volume of the whole solid we sum
  thevolumes of the disks
• Shown as a definite integral

f(x)
a
b
8
w
w
R
R
Axis of Revolution
9
Revolve this rectangle about the x-axis.
Revolve this function about the x-axis.
10
Revolve this rectangle about the x-axis.
Forms a Disk.
Revolve this function about the x-axis.
11
Revolving About y-Axis

• Also possible to revolve a function about the
  y-axis
• Make a disk or a washer to be horizontal
• Consider revolving a parabola about the y-axis
• How to represent the radius?
• What is the thicknessof the disk?

12
Revolving About y-Axis

• Must consider curve asx f(y)
• Radius f(y)
• Slice is dy thick
• Volume of the solid rotatedabout y-axis

13

• Horizontal Axis of Revolution
• Volume V
• Vertical Axis of Revolution
• Volume V

14
Disk Method (to find the volume of a solid of
revolution)
Horizontal Axis of revolution
Vertical Axis of revolution
15
Disk Method (to find the volume of a solid of
revolution)
Horizontal Axis of revolution
radius
width
Vertical Axis of revolution
16
1. Find the volume of the solid formed by
revolving f(x) over about the x-axis.
17
1. Find the volume of the solid formed by
revolving f(x) over about the x-axis.
18
2. Find the volume of the solid formed by
revolving the region formed by f(x) and g(x)
about y 1.
19
2. Find the volume of the solid formed by
revolving the region formed by f(x) and g(x)
about y 1.
Length of Rectangle
20
3. Find the volume of the solid formed by
revolving the region formed by over
about
the y-axis.
21
3. Find the volume of the solid formed by
revolving the region formed by over
about
the y-axis.
22

• The disc method can be extended to cover solids
  of revolution with holes by replacing the
  representative disc with a representative washer.

w
R
r
23
Washers

• Consider the area between two functions rotated
  about the axis
• Now we have a hollow solid
• We will sum the volumes of washers
• As an integral

f(x)
g(x)
a
b
24

• Volume of washer

25
How do you find the volume of the figure formed
by revolving the shaded area about the x-axis?
26
How do you find the volume of the figure formed
by revolving the shaded area about the x-axis?
Outside radius
Inside radius
The volume we want is the difference between the
two.
Revolving this function creates a solid whose
volume is larger than we want.
Revolving this function carves out the part we
dont want.
27
Washer Method (for finding the volume of a solid
of revolution)
28
Washer Method (for finding the volume of a solid
of revolution)
Outside radius
Inside radius
29
1. Find the volume of the solid generated by
revolving the area enclosed by the two functions
about the x- axis.
30
1. Find the volume of the solid generated by
revolving the area enclosed by the two functions
about the x- axis.
Find intersection points first.
Washer Method
31
2. Find the volume of the solid formed by
revolving the area enclosed by the given
functions about the y- axis.
32
2. Find the volume of the solid formed by
revolving the area enclosed by the given
functions about the y- axis.
This region is not always created by the same two
functions.
The change occurs at y 1.
We need to use two integrals to find the volume.
For y in 0,1 we use disk method.
For y in 1,2 we use washer method.
Since were revolving about the y-axis, each
radius must be in terms of y.
(distance from the parabola to the x-axis)
(distance from the parabola to the y-axis)
33
(No Transcript)
34
Volume of Revolution - X

• Find the volume of revolution about the x-axis of
  
• f(x) sin(x) 2 from x 0 to x 2?.

35
Volume of Revolution - X

• Find the volume of revolution about the x-axis of
  
• f(x) sin(x) 2 from x 0 to x 2?

Use the TI to integrate this one!
Did you get 88.826 cubic units?
36
Volumes of Solids with Known Cross Sections

• For cross sections of area A(x) taken
  perpendicular to the x-axis, Volume
• For cross sections of area A(y) taken
  perpendicular to the y-axis,
• Volume

37
Volume of Revolution - X

• Lets look at the cross section or slice. What is
  it?

Its a circle.
What is the radius?
f(x).
What is the area?
A ?r2
A ? (f(x))2
38
Volume of Revolution - X
How wide (thick) is the disc?
dx
The volume of the disk is
V ? (f(x))2dx
How do we add up all the disks from x 1 to x
4?
39
Example 1
Find the volume of the solid whose base is the
region in the xy-plane bounded by the given
curves and whose cross-sections perpendicular to
the x-axis are (a) squares (b) semicircles and
(c) equilateral triangles.
40
Example 1(a) Square Cross Sections

• My Advice???
• Draw the 2-D picture and imagine the
  cross-sectional shape coming out in the third
  dimension!
• Try and think of how to write the area of these
  cross sections using the given function.
• Set up the integral and integrate.

Think SKATE BOARD RAMP!!!
41
Example 1(b) Semicircular Cross Sections
Think CORNICOPIA!!!
42
Example 1(c) Equilateral Triangular Cross
Sections
Think PYRAMID, kinda??
43
Example 2
Find the volume of the solid whose base is the
region in the xy-plane bounded by the given
curves and whose cross-sections perpendicular to
the x-axis are (a) squares (b) semicircles and
(c) equilateral triangles.
44
Example 2 (a) Square Cross Sections
Think PYRAMID!!!
45
Example 2 (b) Semicircular Cross Sections
Think CORNICOPIA AGAIN!!!
46
Example 2 (c) Equilateral Triangular Cross
Sections
Think PYRAMID!!!
47
Example 2 (c) Equilateral Triangular Cross
Sections
Think PYRAMID!!!
48
Theorem The volume of a solid with
cross-section of area A(x) that is
perpendicular to the x-axis is given by
a
b
Finding the volume of a solid with known
cross-section is a 3-step process
49
Theorem The volume of a solid with
cross-section of area A(x) that is
perpendicular to the x-axis is given by
This circle is the base of a 3-D figure coming
out of the screen.
S
a
b
This rectangle is a side of a geometric figure
(a cross-section of the whole).
Finding the volume of a solid with known
cross-section is a 3-step process
Step 1 Find the length (S) of the rectangle
used to create the base of the figure.
Step 2 Find the area A(x) of each cross-section
(in terms of this rectangle).
Step 3 Integrate the area function from the
lower to the upper bound.
Volume of a solid with cross-sections of area
A(y) and perpendicular to the y-axis
d
S
c
50
Find the volume of the solid whose base is
bounded by the circle with
cross-sections perpendicular to the x-axis.
These cross-sections are
a. squares
Step 1 Find S.
Step 2 Find A(x).
Step 3 Integrate the area function.
51
Find the volume of the solid whose base is
bounded by the circle with
cross-sections perpendicular to the x-axis.
These cross-sections are
a. squares
Step 1 Find S.
S
-2
2
Step 2 Find A(x).
S
Step 3 Integrate the area function.
52
Find the volume of the solid whose base is
bounded by the circle with
cross-sections perpendicular to the x-axis.
These cross-sections are
b. equilateral triangles
Step 1 Find S.
Step 2 Find A(x).
Step 3 Integrate.
53
Find the volume of the solid whose base is
bounded by the circle with
cross-sections perpendicular to the x-axis.
These cross-sections are
b. equilateral triangles
Step 1 Find S.
S
-2
2
Step 2 Find A(x).
S
Step 3 Integrate.
54
Volumes by Slicing
Solids with known cross sections Find the volume
of a solid where the base is an ellipse with
semi-major axis of 4 units and semi-minor axis of
3 units if all cross sections perpendicular to
the major axis are semicircles.
55
The natural draft cooling tower shown at left is
about 500 feet high and its shape can be
approximated by the graph of this equation
revolved about the y-axis
The volume can be calculated using the disk
method with a horizontal disk.
56
(No Transcript)
57
(No Transcript)
58
(No Transcript)
59
(No Transcript)
60
(No Transcript)
61
(No Transcript)
62
(No Transcript)
63
(No Transcript)
64
(No Transcript)