PowerPoint Presentation - Solids of Revolution
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Finding Volumes
Disk/Washer/Shell
Chapter 6.2 & 6.3
February 27, 2007
Review of Area: Measuring a length.
Vertical Cut:
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top
bottom
function function
f (x) g(x)
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right
left
function function
f (y) g(y)
Disk Method
Slices are perpendicular to the axis of rotation.
Radius is a function of position on that axis.
Therefore rotating about x axis gives an integral
in x; rotating about y gives an integral in y.
Find the volume of the solid generated by revolving the
3
y
x
region defined by
, y = 8, and x = 0 about the
y-axis.
Bounds?
[0,8]
Length?
x 3 y
y
2
Area?
Volume?
3
8
3`
0
2
y dy
Find the volume of the solid generated by revolving the
region defined by y 2 x 2, and y = 1, about the
line y = 1
[-1,1]
Bounds?
2
(2
x
) 1
Length?
Area?
1 x
1
2 2
Volume? 1 x
1
dx
2 2
What if there is a “gap” between the axis of rotation and
the function?
Solids of Revolution:
We determined that a cut perpendicular to
the axis of rotation will either form a
disk (region touches axis of rotation
(AOR)) or
a washer (there is a gap between the
region and the AOR)
Revolved around the line y = 1, the region
forms a disk
However when revolved around the x-axis,
there is a “gap” between the region and
the x-axis. (when we draw the radius, the
radius intersects the region twice.)
Area of a Washer
Area Area
of
of
Outer Inner
R
r
R2 r 2
R 2 r 2
Note: Both R and r are measured
from the axis of rotation.
Find the volume of the solid generated by revolving the
region defined by y 2 x 2, and y = 1, about the
x-axis using planar slices perpendicular to the AOR.
[-1,1]
Bounds?
Outside Radius?
1
Inside Radius?
Area? 2 x
1
Volume?
1
2 x
2 2
1 dx
2
2 x2
1
2 2
2
Find the volume of the solid generated by revolving the
region defined by y 2 x 2, and y = 1, about the
line y=-1.
Bounds?
[-1,1]
Outside Radius? 2 x 2 1
Inside Radius?
Area?
3 x
2
2 2
2
3 x 2 dx
1
Volume?
1 1
1
2 2
2
Let R be the region in the x-y plane bounded by
4
1
y , y , and x 2
x
4
Set up the integral for the volume obtained by rotating R
about the x-axis using planar slices perpendicular to the
axis of rotation.
4
1
y , y , and x 2
x
4
Notice the gap:
Outside Radius ( R ):
4
x
Inside Radius ( r ):
1
4
2
Area:
2
4
1
2 x 4 dx
16
Volume:
2
4
1
x
4
2
Let R be the region in the x-y plane bounded by
y 2x 2 and y 3x 1
Set up the integral for the volume of the solid obtained
by rotating R about the x-axis, using planar slices
perpendicular to the axis of rotation.
Notice the gap:
y 2x 2 and y 3x 1
Outside Radius ( R ): 3x 1
Inside Radius ( r ):
2x 2
Area: 3x 1 2x
2
3x 1
1
2
Volume:
1
2
dx
2x
2 2
2 2
Find the volume of the solid generated by revolving the
region defined by y x , x = 3 and the x-axis about
the x-axis.
[0,3]
Bounds?
Length? (radius) y x
Area?
x
2
3
Volume?
xdx
0
Note in the disk/washer methods, the focus in on the
radius (perpendicular to the axis of rotation) and the
shape it forms. We can also look at a slice that is
parallel to the axis of rotation.
Note in the disk/washer methods, the focus in on the
radius (perpendicular to the axis of rotation) and the
shape it forms. We can also look at a slice that is
parallel to the axis of rotation.
2 r
Length
of slice
length
Area: 2 r of
slice
radius length
y
d
Volume = 2 from of
x
a
AOR slice
b
Slice is PARALLEL to the AOR
Using y x on the interval [0,2] revolving around
the x-axis using planar slices PARALLEL to the AOR,
we find the volume:
Length of slice?
Radius?
2 y2
y
Area? 2 y 2 y
2
Volume?
2
2
2
y
2
y
dy
0
4
1
y , y , and x 2
x
4
Back to example:
Find volume of the solid generated by revolving the
region about the y-axis using cylindrical slices
4 1
Length of slice ( h ):
x 4
Radius ( r ):
Area:
4
2 2 x x
16
Volume:
1
dx
4
x
4
2 x
x
1
4
Find the volume of the solid generated by revolving the
region: y 2x 2 and y 3x 1
about the y-axis, using cylindrical slices.
Length of slice ( h ):
3x 1 2x 2
Inside Radius ( r ):
x
Area: 2 x 3x 1 2x
1
Volume:
2
2
x
3x
1
2x
dx
1
2
2
Try:
Set up an integral integrating with respect
to y to find the volume of the solid of
revolution obtained when the region
bounded by the graphs of y = x2 and y = 0
and x = 2 is rotated around
a) the y-axis
b) the line y = 4