Volume by Integral Calculus Techniques.ppt

VOLUME BY INTEGRATION
(SOLIDS OF REVOLUTION)

•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 = r2
h

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
h
r
V 2


  dx
x
f
V
2
0
)
( 

To find the volume of the entire solid  


b
a
dx
x
f
V
2
)
(


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 elements
are perpendicular to and touches the
axis of revolution, thus we use the disk
Method.

We use to find the area of the strip.
h
r
V 2


 
   
   
 
6
0
3
6
0
2
6
0
2
6
0
2
2
3
y
6
4
V
dy
y
6
4
V
dy
y
6
2
1
V
y
6
2
1
x
6;
-2x
y
if
,
dy
x
V
dy
x
dV


































   
 
 
 
units
cu.
18
V
6
36
12
V
0
6
6
6
12
V 3
3













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 = dx
y1 = g(x) y2 = f(x)
 
 

 




b
a
dx
y
y
V
dx
y
y
dV
2
2
2
1
2
2
2
1

 Since )
(
1 x
f
y 
)
(
2 x
g
y 
   
 




 


b
a
2
2
dx
x
f
x
g
V
and
r
R

Equations (7) – (8) (p. 426)
Figure 6.2.14

y
4
x2

 0
1

x
.
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 )
x
0
x2
= 4-y x-1=0
r =1- 0 = 1
y


(-2, 0 )
By horizontal stripping, the elements are
perpendicular but not touching the axis
of revolution, thus we use the
Ring or Washer Method.

  h
r
R
V 2
2


 2
)
4
(
1 y
R 



2
)
y
4
1
( 

 1

r
 dy
1
)
y
4
1
(
dV 2
2





dy
)
1
y
4
y
4
2
1
(
V
4
0
 






 dy
y
4
2
)
y
4
(
4
0
 




0
4
2
/
3
)
y
4
(
2
2
)
y
4
( 2
/
3
2





 


























 2
/
3
2
2
/
3
2
)
0
4
(
3
4
)
0
4
(
2
1
)
(
)
4
4
(
3
4
)
4
4
(
2
1

 

3
56
cu. 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
h
rt
Vshell 
 
2

Example:
Find the volume of the solid generated by revolving the second
quadrant region bounded by the curve
about .
y
4
x2

 0
1

x
Using vertical stripping, the
elements parallel to the axis
of revolution, thus we use the
shell method.
Shell Method: rht
V 
2

dx
t
y
h
x
r



1

 
 
 
   
 
 







































2
0
3
2
2
0
2
2
2
2
2
0
2
0
0
2
dx
x
x
x
4
4
2
dx
x
4
x
x
4
2
x
-
4
y
;
y
4
x
but
dx
xy
y
2
ydx
x
1
2
ydx
x
1
2
V
ydx
)
x
1
(
2
dV
       
2
2 3 4
0
2 3 4
2 4 4
2 3 4
1 1
2 4 2 2 2 2 2
3 4
8
2 8 8 4
3
x x x
V x




 
    
 
 
 
       
 
 
 
    
 
 
8
2 12
3
36 8 56
2 cu. units
3 3



 
  
 
 
 
 
 
 
 

HOMEWORK
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-axis
2.y = 6x – x2
, y = 0; about x-axis
3.y2
= 4x, x = 4; about x = 4
4.y = x2
, y2
= x; about x = -1
5.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
1
4 4
x
x
y 

2. y-axis, about x=2

Volume by Integral Calculus Techniques.ppt

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