Solids of Revolution


The Bell!
We start with a known planar shape and rotate that shape about
a line resulting in a three dimensional shape known as a solid of
revolution. When this solid of revolution takes on a non-regular
shape, we can use integration to compute the volume.
For example……
Area:
! ( f (x))
2
Volume:
b
" ! ( f (x))
2
dx
a
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.
Bounds?
Length? (radius)
Area?
Volume?
Solids of Revolution
For solids of revolution, cross sections are circles, so we
can use the formula
" ! ( r(h))
b
Volume =
a
2
dh
Usually, the only difficult part is determining r(h).
A good sketch is a big help.
4
Aside: Sketching Revolutions
1. Sketch the curve; determine the region.
2. Sketch the reflection over the axis.
Example
Rotate region between x = sin(y), 0 ≤ y ≤ π, and the
y axis about the y axis.
3. Sketch in a few “revolution” lines.
x = sin(y)
Example
Revolve the region under the curve y = 3e–x, for 0 ≤ x ≤ 1,
about the x axis.
Remember for this 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.
5
Find the volume of the solid generated by revolving the
2
region defined by y = x , on the interval [1,2] about the
x-axis.
Example
Rotate y = x2, from x = 0 to x = 4, about the x-axis.
Find r(x): r(x) =
Bounds?
Length? (radius)
Area?
Volume?
Find the volume of the solid generated by revolving the
3
region defined by y = x , y = 8, and x = 0 about the
y-axis.
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?
Bounds?
Length?
Length?
Area?
Area?
Volume?
Volume?
6
*Find the volume of the solid generated by revolving the
region defined by y = 2 ! x 2, and y = 1, about the
x-axis.
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?
Bounds?
Outside Radius?
Outside Radius?
Inside Radius?
Inside Radius?
Area?
Area?
Volume?
Volume?
Solids of Revolution
For solids of revolution, cross sections are circles,
If there is a gap between the function and the axis of
rotation, we have a washer and use:
Volume =
#
b
a
(
)
! ( R(h)) " (r(h))2 dh
2
If there is NO gap, we have a disk and use:
Volume =
" ! ( r(h))
b
a
2
dh
7
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.)
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.
Area of a Washer
! Area $ ! Area $
# of
& ' # of &
#
& #
&
" Outer % " Inner %
R
r
Bounds?
Outside Radius?
! ( R2 ) " ! ( r 2 )
Inside Radius?
! #$( R 2 ) " ( r 2 ) %&
Area?
Note: Both R and r are measured
from the axis of rotation.
Volume?
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.
Let R be the region in the x-y plane bounded by
y=
4
1
,!!y = ,!!and!!x = 2
x
4
Bounds?
Set up the integral for the volume obtained by
rotating R about the x-axis using planar slices
perpendicular to the axis of rotation.
Outside Radius?
Notice the gap:
Inside Radius?
Outside Radius ( R ):
Area?
Inside Radius ( r ):
Volume?
Area:
Volume:
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 obtained by
rotating R about the x-axis using planar slices
perpendicular to the axis of rotation.
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.
Bounds?
Length? (radius)
Notice the gap:
Area?
Outside Radius ( R ):
Inside Radius ( r ):
Volume?
Area:
Volume:
3
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 &
b
Volume =
" radius % " length %
' $ of
'd " y%
'$
' $# x '&
# AOR & # slice &
( 2! $$ from
a
Using y = x on the interval [0,2] revolving around
the x-axis using planar slices PARALLEL to the AOR,
we find the volume:
Radius?
Length of slice?
Area?
Volume?
Slice is PARALLEL to the AOR
4
4
1
y = ,!!y = ,!!and!!x = 2
Back to example:
x
4
Find volume of the solid generated by revolving the
region about the y-axis using cylindrical slices
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 ):
Volume:
Length of slice ( h ):
Radius ( r ):
Inside Radius ( r ):
Area:
Area:
Volume:
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