Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution...

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Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids

Transcript of Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution...

Page 1: 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

Examples: Classify the solids

Page 2: 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

Page 3: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

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

Page 4: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

VOLUMES

1 The Disk Method

Strip with small width generate a disk after the rotation

Page 5: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

VOLUMES

1 The Disk Method

Several disks with different radius r

Page 6: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

VOLUMES

xrV 2

Page 7: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

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

Page 8: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

VOLUMES

Page 9: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

VOLUMES

Page 10: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

Volumes Using Cross-Sections

Volumes Using Cross-Sections

Sec(6.1)

The DiskMethod

The WasherMethod

Page 11: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

Volumes Using Cross-Sections

Volumes Using Cross-Sections

Sec(6.1)

The DiskMethod

The WasherMethod

Examples: Classify

Page 12: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

Volumes Using Cross-Sections

Volumes Using Cross-Sections

Sec(6.1)

The DiskMethod

The WasherMethod

Examples: Classify

Page 13: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

VOLUMES

Volume = Area of the base X height

1r

2r

washer

x

r

x

disk

xrV 2

xrxrV 21

22

xrrV 21

22

Page 14: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

VOLUMES

xrrV 21

22

Page 15: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

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

Page 16: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

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

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VOLUMES T-102

),( xex

),(1

1

xx

)0,(x

Page 18: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

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

Page 19: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

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)

Page 20: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

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

Page 21: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

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)

Page 22: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

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

Page 23: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

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

Page 24: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

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

Page 25: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

parallel to x-axis

VOLUMES

dxV

dyV

parallel to y-axis

dxV

dyV

SHELLS

Cross-Section

Page 26: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

VOLUMES BY CYLINDRICAL SHELLS

T-131

Remark: before you start solving the problem, read the choices to figure out which method you use

Page 27: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

T-111

Page 28: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

VOLUMES

T-102

Page 29: 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

Page 30: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

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:

Page 32: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

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

Page 33: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

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

Page 34: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

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

Page 35: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

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

Page 36: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

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

Page 37: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

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

Page 38: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

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

Page 39: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

T-102

VOLUMES

Page 40: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

T-122

VOLUMES

Page 41: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

T-092

VOLUMES

Page 42: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

VOLUMES

Page 43: Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

T-132

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T-132