8.2 Area of a Surface of RevolutionDefinition:

If the graph of a continuous function is revolved about a line, the resulting surface is called a surface of revolution

Axis of Revolution

L

1r2r

)(2

1 radius average

segment line oflength where2

21 rrr

LrLS

Surface Area of a Frustum of a Cone:

Surface Area:

ydsS 2 If the curve is rotated about the x-axis

xdsS 2 If the curve is rotated about the y-axis

Note:

)( when 12

xfydxdx

dyds

Use

or

)( when 12

ygxdydy

dxds

Examples:Find the area of the surface obtained by rotating the curve about the x-axis:

1) 80 ,442 xxy

Solutions:

21053

16840

3

43

2

24

4

42

20 when :LimitLower

68 when :LimitUpper

4

4

411

22

4

1

1 Use

4

4

44 Now,

2 have We

2/32/3

6

2

2/322/16

2

26

2

2

222

2

2

2

ydyyydyy

yS

yx

yx

yy

dy

dxyy

dy

dx

dydy

dxds

yx

xy

ydsS

2) rotating about the x-axis3/0 ,cos xxy

Solutions:

2

37ln

4

21

2

3

2

7ln

2

3

2

7)tanln(sec

2

1tansec

2

12

Partsby n Integratio sec2sectan12

dsec

2/2/ ,tanlet

12

0 :LimitLower cos 2

3 :Limitper Up sinlet

sin1cos2

sin1sin11 Use

2 have We

2/3tan

0

32/3tan

0

22/3tan

0

2

2

2/3

0

2

3/

0

2

222

Arc

ArcArcddS

dv

v

duuS

uxdxdu

uxu

dxxxS

dxxdxxdxdx

dyds

ydsS

Find the area of the surface obtained by rotating the curve about the y-axis:

3) 10 ,2 2 yyyx

2222

122

2

1

2

121

2

1

2

1222

2

1

1 where

2

have We

1

0

1

0

1

0 22

22

222

2

22

2

2/12

2

ydydyyy

yyS

yyyy

yyy

dy

dx

yy

y

dy

dx

yy

yyyy

dy

dx

dydy

dxds

xdsS

Solutions: