TOPIC

APPLICATIONSVOLUME BY INTEGRATION

•define what a solid of revolution is

•decide which method will best determine the volume of the solid

•apply the different integration formulas.

OBJECTIVE

DEFINITION

A solid of revolution is the figure formed when a plane region is revolved about a fixed line. The fixed line is called the axis of revolution. For short, we shall refer to the fixed line as axis.

The volume of a solid of revolution may be using the following methods: DISK, RING and SHELL METHOD

This method is used when the element (representative strip) is perpendicular to and touching the axis. Meaning, the axis is part of the boundary of the plane area. When the strip is revolved about the axis of rotation a DISK is generated.

A. DISK METHOD: V = r2h

h = dx

y

dx

x = a

f(x) - 0

x = b

y = f(x)

x

= r

The solid formed by revolving the strip is a cylinder whose volume is

hrV 2 dxxfV 20)(

To find the volume of the entire solid b

a

dxxfV 2)(

Equation Volume by disks

Example:Find the volume of the solid generated by revolving the region bounded by the line y = 6 – 2x and the coordinate axes about the y-axis.

r =x

(x , y)

h = dy

(0 , 6 )

x

y

0(3,0)

By horizontal stripping, the elementsare perpendicular to and touches the axis of revolution, thus we use the disk Method.

We use to find the area of the strip. hrV 2

6

0

3

6

0

2

6

0

2

6

0

2

2

3

y6

4V

dyy64

V

dyy62

1V

y62

1 x6;-2x yif , dyxV

dyxdV

units cu. 18V

63612

V

066612

V 33

Ring or Washer method is used when the element (or representative strip) is perpendicular to but not touching the axis. Since the axis is not a part of the boundary of the plane area, the strip when revolved about the axis generates a ring or washer.

B. RING OR WASHER METHOD: V = (R2 – r2)h

(x1 , y1)

(x2 , y2)

x = a

x = b

dx

h = dxy1 = g(x) y2 = f(x)

b

adxyyV

dxyydV

2

2

2

1

2

2

2

1

Since )(1 xfy )(2 xgy

b

a

22dxxfxgV

and

rR

Figure 6.2.15 (p. 427)

Equations (7) – (8) (p. 426)

Figure 6.2.14

y4x2 01x.

Example:Find the volume of solid generated by revolving the second quadrant region bounded by the curve about .

R =1-x

h =d y

(0 , 4 )

x0

x2 = 4-y x-1=0

r =1- 0 = 1

y

(-2, 0 )

By horizontal stripping, the elements areperpendicular but not touching the axis of revolution, thus we use theRing or Washer Method.

hrRV 22

2)4(1 yR 2)y41( 1r

dy1)y41(dV 22

dy)1y4y421(V4

0

dyy42)y4(4

0

0

4

2/3

)y4(2

2

)y4( 2/32

2/322/32 )04(

3

4)04(

2

1)()44(

3

4)44(

2

1

3

56cu. units.

The method is used when the element (or representative strip) is parallel to the axis of revolution. When this strip is revolved about the axis, the solid formed is of cylindrical form.

C. SHELL METHOD

hrtVshell 2

Example:Find the volume of the solid generated by revolving the second quadrant region bounded by the curve about .

y4x2 01x

Using vertical stripping, the elements parallel to the axis of revolution, thus we use the shell method.

Shell Method: rhtV 2

dxt

yh

xr

1

2

032

2

022

222

0

2

0

0

2

dxxxx442

dxx4xx42

x-4 y;y4 xbut dxxyy2

ydxx12

ydxx12V

ydx)x1(2dV

22 3 4

0

2 3 4

2 4 42 3 4

1 12 4 2 2 2 2 2

3 4

82 8 8 4

3

x x xV x

82 12

3

36 8 562 cu. units

3 3

HOMEWORK #8

A. Using disk or ring method, find the volume generated by revolving about the indicated axis the areas bounded by the following curves:

1.y = x3, y = 0, x = 2; about x-axis2.y = 6x – x2, y = 0; about x-axis3.y2 = 4x, x = 4; about x = 44.y = x2, y2 = x; about x = -15.y = x2 – x, y = 3 – x2; about y = 4

B. Using cylindrical shell method, find the volume generated by revolving about the indicated axis the areas bounded by the following curves:

1.y = 3x – x2, the y-axis, y = 2; about y-axis

3. y = x3, x = y3; about x-axis

,8

14 4xxy 2. y-axis, about x=2