AP CALCULUS - AB
Section Number:
LECTURE NOTES
Topics: Volume: The Disk & Washer Methods
MR. RECORD
Day: 1 of 2
7.2
Area is only one of the many applications of the definite integral. We can also use it to find the volume of a threedimensional solid. The solids which you will focus on in this unit are those with cross sections that are all similar.
w
w
R( x )
R( x )
A 2-dimensional figure ………..which, when rotated about the x-axis becomes a…….3-dimensional figure
Recall: The formula for the volume of a cylinder is V   r h . 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:
2
The Disk Method
To find the volume of a solid of revolution with the disk method, use one of the following:
Horizontal Axis of Revolution
b
Vertical Axis of Revolution
d
Volume  V    R(x) dx
Volume  V    R(y) dy
2
a
2
c
x
d
y
R( x )
a
b
c
R(y)
Example 1: Using the Disk Method.
Find the volume of the solid formed by revolving the region bounded by the graphs of f (x)  sin x
and the x-axis about the x-axis (0  x   ) .
Sketch the graph and draw a representative rectangle.

y
x




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 f (x)  2  x 2
and g(x)  1 about the line y  1
Sketch the graph and draw a representative rectangle.
y



x





The Washer Method
To find the volume of a solid of revolution with the washer method, use one of the following:
Horizontal Axis of Revolution
b

Vertical Axis of Revolution

d
Volume  V    R(x)   r (x) dx
2
2


Volume  V    R(y)   r (y) dy
a
2
2
c
R(y)
d
R( x )
r (x)
a
b
c
r (x)
Example 3: Using the Washer Method.
Find the volume of the solid formed by revolving the region bounded by the graphs of f (x)  x
and y  x 2 about the x-axis.
Sketch the graph and draw a representative rectangle.
y


x



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
y  x 2  1, y  0, x  0, and x  1 about the y-axis.
Sketch the graph and draw a representative rectangle.
y



x




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)

AP CALCULUS
Section Number:
LECTURE NOTES
Topics: Volume: Using KNOWN CROSS SECTIONS
MR. RECORD
Day: 2 of 2
7.2
Solids With Known Cross Sections
Volumes of Solids with Known Cross Sections
1. For cross sections of area A(x) taken perpendicular to the x-axis.
b
Volume   A( x) dx
a
2. For cross sections of area A(y) taken perpendicular to the y-axis.
d
Volume   A(y) dy
c
Example 5: Triangular Cross Sections.
Find the volume of the solid in which the base is the region bounded by the graphs of
x
x
f (x)  1  , g(x)  1  , and x  0.
2
2
a. The cross sections perpendicular to the x-axis are squares.
y


x




b.
The cross sections perpendicular to the x-axis are semi-circles.
c.
The cross sections perpendicular to the x-axis are equilateral triangles.


