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

Jonathan Mitchell

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