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2017 VolumesDisk and Washer Methods.docx
2017 VolumesDisk and Washer Methods.pdf
AP CALCULUS - AB
NOTES and EXAMPLES
DeMalade
Topics: Volume: The Disk & Washer Methods
Area is only one of the many applications of the definite
integral. We can also use it to find the volume of a
three-dimensional solid. The solids which you will focus on in this
unit are those with cross sections that are all circular.
(w) (w)
A 2-dimensional figure ………..which, when rotated about the x-axis
becomes a…….3-dimensional figure – a cyclinder.
Recall: The formula for the volume of a cylinder is . A disk is
basically a very short cylinder where the height, h is actually the
w value. The radius retains its actual meaning. The pops out in
front and you get the following formula below:
The Disk Method
To find the volume of a solid of revolution with the disk
method, use one of the following:
Horizontal Axis of RevolutionVertical Axis of Revolution
(d)
(c)
a b
Example 1: Using the Disk Method.
Find the volume of the solid formed by revolving the region
bounded by the graphs of
and the x-axis about the x-axis .
Sketch the graph and draw a representative rectangle.
Example 2: Revolving About a Line That Is Not A Coordinate
Axis.
Find the volume of the solid formed by revolving the region
bounded by the graphs of
and about the line
Sketch the graph and draw a representative rectangle.
The Washer Method
To find the volume of a solid of revolution with the washer
method, use one of the following:
Horizontal Axis of RevolutionVertical Axis of Revolution
(d)
(c)
a b
Example 3: Using the Washer Method.
Find the volume of the solid formed by revolving the region
bounded by the graphs of
and about the x-axis.
Sketch the graph and draw a representative rectangle.
Example 4: Integrating with Respect to y, Two Integral Case.
Find the volume of the solid formed by revolving the region
bounded by the graphs of about the y-axis.
Sketch the graph and draw a representative rectangle.
Think of the above example in some other ways.
What would your integration set-up look like if you were to
revolve the region about
(a) the line x =–1?(b) the line y = -2?(c) the line x = 2?
Advice
1. Draw a representative rectangle that begins from the axis of
revolution to the farthest boundary.
(This distance will represent R)
2. Draw a representative rectangle that begins from the axis of
revolution to the nearest boundary.
(This distance will represent r)
? Is the “slice” solid? i.e. touches the axis of revolution at
all times? Use disc method
? Does the “slice” have a hole in it? i.e. does the original
graph touch the axis of revolution at all times?
Use the washer method
AP CALCULUS
NOTES and EXAMPLES
DeMalade
Topics: Volume: Using Known Cross Sections
Solids With Known Cross Sections
(Volumes of Solids with Known Cross Sections1. For cross
sections of area A(x) taken perpendicular to the x-axis. 2. For
cross sections of area A(y) taken perpendicular to the y-axis.
)
Example 5: Triangular Cross Sections.
Find the volume of the solid in which the base is the region
bounded by the graphs of
a. The cross sections perpendicular to the x-axis are
squares.
b. The cross sections perpendicular to the x-axis are
semi-circles.
c. The cross sections perpendicular to the x-axis are
equilateral triangles.
2
Vrh
p
=
p
[
]
2
Volume ()
b
a
VRxdx
p
==
ò
[
]
2
Volume ()
d
c
VRydy
p
==
ò
x
D
y
D
()
Rx
()
Ry
()sin
fxx
=
(0)
x
p
££
x
y
2
()2
fxx
=-
()1
gx
=
1
y
=
x
y
[
]
[
]
(
)
22
Volume ()()
b
a
VRxrxdx
p
==-
ò
[
]
[
]
(
)
22
Volume ()()
d
c
VRyrydy
p
==-
ò
()
Ry
()
Rx
()
rx
()
rx
()
fxx
=
2
yx
=
x
y
2
1,0,0, and 1
yxyxx
=+===
x
y
Volume ()
b
a
Axdx
=
ò
()
Rx
Volume ()
d
c
Aydy
=
ò
()1, g(x)1,and 0.
22
xx
fxx
=-=-+=
x
y
()
Rx
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