Volumes Using Cross-Sections
Solids of Revolution
Solids
Solids not generated by
Revolution
Examples: Classify the solids
Volumes Using Cross-Sections
Solids of Revolution
Solids
Solids not generated by
Revolution
Volumes Using Cross-Sections
Solids of Revolution
Volumes Using Cross-Sections
Volumes Using Cylindrical Shells
Sec(6.2)Sec(6.1)
The DiskMethod
The WasherMethod
VOLUMES
1 The Disk Method
Strip with small width generate a disk after the rotation
VOLUMES
1 The Disk Method
Several disks with different radius r
VOLUMES
xrV 2
VOLUMES
1 Disk cross-section x
step1 Graph and Identify the region
step2 Draw a line (L) perpendicular to the rotating line at the point x
step4 Find the radius r of the circe in terms of x
step5Now the cross section Area is
2 rA
step6Specify the values of x
bxa
step7The volume is given by
b
adxxAV )(
1p
Intersection point between L, rotating axis
Intersection point between L, curve
2p
xr
1
0
2)( dxV x 2
1
step3 Rotate this line. A circle is generated
)0,(x
),( xx
VOLUMES
VOLUMES
Volumes Using Cross-Sections
Volumes Using Cross-Sections
Sec(6.1)
The DiskMethod
The WasherMethod
Volumes Using Cross-Sections
Volumes Using Cross-Sections
Sec(6.1)
The DiskMethod
The WasherMethod
Examples: Classify
Volumes Using Cross-Sections
Volumes Using Cross-Sections
Sec(6.1)
The DiskMethod
The WasherMethod
Examples: Classify
VOLUMES
Volume = Area of the base X height
1r
2r
washer
x
r
x
disk
xrV 2
xrxrV 21
22
xrrV 21
22
VOLUMES
xrrV 21
22
n
ii
nxxAV
1
*)(lim
b
adxxAV )(
If the cross-section is a washer ,we find the inner radius and outer radius
22 )()( inout rrA
VOLUMES
2 The washer Method
VOLUMES
step1 Graph and Identify the region
step2 Draw a line perpendicular to the rotating line at the point x
step4 Find the radius r(out) r(in) of the washer in terms of x
step5 Now the cross section Area is
)( 22inout rrA
step6 Specify the values of x bxa
step7The volume is given by b
adxxAV )(
2 The washer Method
1p
Intersec pt between L, rotation axis
Intersection point between L, boundary
2p
3p
Intersection point between L, boundaryxy
2xy
step3 Rotate this line. Two circles created
)0,(x
),( xx
),( 2xx
x
0
02
xr
xr
out
in
)1,1(
)0,0(
1
0
42 dxxx
VOLUMES T-102
),( xex
),(1
1
xx
)0,(x
Example:
VOLUMES
Find the volume of the solid obtained by rotating the region enclosed by the curves y=x and y=x^2 about the line y=2 . Find the volume of the resulting solid.
)1,1(
xy
2xy
2y
x
)2,(x
),( xx
),( 2xx
in
out
r
r
x
x
2
2 2
1
0
222 )2()2( dxxxV
VOLUMES
n
ii
nyyAV
1
*)(lim
d
cdyyAV )(
If the cross-section is a disk, we find the radius of the disk (in terms of y ) and use
2)(radiusA
3 The Disk Method (about y-axis)
VOLUMES
The Disk Method (about y-axis)
step1 Graph and Identify the region
step2 Draw a line (L) perpendicular to the rotating line at the point y
step4 Find the radius r of the circe in terms of y
step5Now the cross section Area is
2 rA
step6Specify the values of y
dyc
step7The volume is given by
d
cdyyAV )(
step3 Rotate this line. A circle is generated
VOLUMES
4Example: The region enclosed by the curves y=x and y=x^2 is rotated
about the line x= -1 . Find the volume of the resulting solid.
)1,1(
xy
2xy
1x
y),1( y ),( yy),( yy
in
out
r
r
1
1
y
y 1
0
22 )1()1( dxyyV
The Washer Method (about y-axis or parallel)
VOLUMES
4 washer cross-section y
1x
yx yx
step1 Graph and Identify the region
step2 Draw a line perpendicular to the rotating line at the point y
step4 Find the radius r(out) r(in) of the washer in terms of y
step5 Now the cross section Area is
)( 22inout rrA
step6 Specify the values of y dyc
step7The volume is given by d
cdyyAV )(
step3 Rotate this line. Two circles created
solids of revolution
VOLUMES
SUMMARY:The solids in all previous examples are all called solids of revolution because they are obtained by revolving a region about a line.
b
adxxAV )(Rotated by a line
parallel to x-axis ( y=c)
d
cdyyAV )(Rotated by a line
parallel to y-axis ( x=c)
NOTE: The cross section is perpendicular to the rotating line
solids of revolution
Cross-section is DISK
Cross—section is WASHER
2)(RA
22 )()( rRA
VOLUMES BY CYLINDRICAL SHELLS
Remarks
CYLINDRICAL SHELLS(6.2)
rotating lineParallel to x-axis
rotating lineParallel to y-axis
dy
dx
Remarks
Using Cross-Section(6.1)
rotating lineParallel to x-axis
rotating lineParallel to y-axis dy
dx
Cross-section is DISK
Cross—section is WASHER
2)(rA 22 )()( inout rrA
SHELL Method
rhA 2
parallel to x-axis
VOLUMES
dxV
dyV
parallel to y-axis
dxV
dyV
SHELLS
Cross-Section
VOLUMES BY CYLINDRICAL SHELLS
T-131
Remark: before you start solving the problem, read the choices to figure out which method you use
T-111
VOLUMES
T-102
Volumes Using Cross-Sections
Solids of Revolution
Solids
Solids not generated by
Revolution
Volumes Using Cross-Sections
The base of a solid is bounded by the curve y = x /2 and the line y =2. If the cross-sections of the solid perpendicular to the y-axis are squares, then find the volume of the solid
Example:2
Base:
22
1 xy
2y
y
x
is bounded by the curveand the line y =2
22
1 xy If the cross-sections of the solid perpendicular to the y-axis are squares
Cross-sections:
VOLUMES
The base of a solid is bounded by the curve and the line x =9. If the cross-sections of the solid perpendicular to the x-axis are semicircle, then find the volume of the solid
Example:
Base: 9x
y x
is bounded by the curveand the line x =9
xy
If the cross-sections of the solid perpendicular to the x-axis are semicircle
Cross-sections:
xy
xy
VOLUMES
The base of a solid is bounded by the curve and the line x =9. If the cross-sections of the solid perpendicular to the x-axis are semicircle, then find the volume of the solid
Example:
Base:9x
y x
is bounded by the curveand the line x =9
xy
If the cross-sections of the solid perpendicular to the x-axis are semicircle
Cross-sections:
xy
xy
VOLUMES
The base of a solid is bounded by the curve and the line x =9. If the cross-sections of the solid perpendicular to the x-axis are semicircle, then find the volume of the solid
Example:
Base:9x
y x
is bounded by the curveand the line x =9
xy
If the cross-sections of the solid perpendicular to the x-axis are semicircle
Cross-sections:
xy
xy
VOLUMES
The base of a solid is bounded by the curve and the line y = 0 from x=0 to x=pi. If the cross-sections of the solid perpendicular to the x-axis are equilatral triangles.
Example:
Base:
y x
is bounded by the curveand the line y =0
xy sin
If the cross-sections of the solid perpendicular to the x-axis are semicircle
Cross-sections:
xy sin
xy sin
VOLUMES
The base of a solid is bounded by the curve and the line y = 0 from x=0 to x=pi. If the cross-sections of the solid perpendicular to the x-axis are equilatral triangles.
Example:
Base:
y x
is bounded by the curveand the line y =0
xy sin
If the cross-sections of the solid perpendicular to the x-axis are equilatral triangles
Cross-sections:
xy sin
xy sin
Volumes Using Cross-Sections
The base of a solid is bounded by the curve y = x /2 and the line y =2. If the cross-sections of the solid perpendicular to the y-axis are squares, then find the volume of the solid
Example:2
22
1 xy
y
x
y
If the cross-sections of the solid perpendicular to the y-axis are squares
Cross-sections:
),( 2 yy
),( 2 yy
yS 22
y
yA
8
)2(4
2
08ydyV
step1 Graph and Identify the region ( graph with an angle)
step2Draw a line (L) perpendicular to the x-axis (or y-axis) at the point x (or y), (as given in the problem)
step4Cross-section type:Square S = side lengthSemicircle S = diameterEquilatral S = side length
step6bxa
step7The volume is given by
b
adxxAV )(
step3Find the length (S)of the segment from the two intersection points with the boundary
step4Cross-section type:
Square
Semicircle
Equilatral
2SA
22
1 SA 2
8
3 SA
Specify the values of x
VOLUMES
The base of a solid is bounded by the curve and the line x =9. If the cross-sections of the solid perpendicular to the x-axis are semicircle, then find the volume of the solid
Example:
y x
If the cross-sections of the solid perpendicular to the x-axis are semicircle
Cross-sections:
xy step1 Graph and Identify the region (
graph with an angle)
step2Draw a line (L) perpendicular to the x-axis (or y-axis) at the point x (or y), (as given in the problem)
step4Cross-section type:Square S = side lengthSemicircle S = diameterEquilatral S = side length
step6Specify the values of x bxa
step7The volume is given by
b
adxxAV )(
step3Find the length (S)of the segment from the two intersection points with the boundary
step4Cross-section type:
Square
Semicircle
Equilatral
2SA
22
1 SA 2
8
3 SA
xy
),( xx
)0,(x
xS
x
A x
4
22
1
)(
9
0 4dxx
V
T-102
VOLUMES
T-122
VOLUMES
T-092
VOLUMES
VOLUMES
T-132
T-132