(Application of definite integral) Volume using Cylindrical Shells Method, Why we can't use disk and washer method,Comparison

2. Volume using Cylindrical
Shells
Outline:
 Statement
 Volume Calculation for two Shells
 Volume Calculation for rotation along y-axis
and x-axis.
 Examples
 Comparison

3. Cylindrical Shells Method
 Statement:
 Let f be the continuous and nonnegative on [a, b],
and let R be the region that is bounded above by
y=f(x), below by the x-axis, and on the sides by the
lines x= a and x= b. Find the volume V of the solid of
revolution “S” that is generated by revolving the
region R about the y- axis.

4. Why Cylindrical Method Used?
Some volume problems are
very
difficult to handle by the using
the methods of disk and washer
by using cross sectional Areas.

5. Cylindrical Shells Method
Fortunately, there is a method—
the
method of cylindrical shells—that is
easier to use in such a case.

6. • Cylindrical Shells Method
The figure shows a Cylindrical Shell
with inner radius r1, outer radius r2
and a height h.

7. • Cylindrical Shells Method
Its volume V is calculated by
subtracting
the volume V1 of the inner cylinder from
the volume of the outer cylinder V2 .

8. Cylindrical Shells Method
Thus, we have;
2 1
2 2
2 1
2 2
2 1
2 1 2 1
2 1
2 1
( )
( )( )
2 ( )
2
V V V
r h r h
r r h
r r r r h
r r
h r r
 



 
 
 
  

 

9. Cylindrical Shells Method
 Let ∆r = r2 – r1 (thickness of the shell)
and
(average radius of the
shell).
 Then, this formula for the volume of a
cylindrical shell becomes:
2
V rh r

 
 
1
2 1
2
r r r
 

10. Cylindrical Shells Method
The equation can be remembered as:
V = [circumference] [height]
[thickness]
2
V rh r

 

11.  Now, let S be the
solid
obtained by rotating
about the y-axis the
region bounded by
y = f(x) [where f(x) ≥
0],
y = 0, x = a and x =
b,
where b > a ≥ 0.
CYLINDRICAL SHELLS METHOD

12. Divide the interval [a, b] into n subintervals
[xi - 1, xi ] of equal width and let be
the midpoint of the i th subinterval.
CYLINDRICAL SHELLS METHOD
i
x x


13.  The rectangle with
base [xi - 1, xi ] and
height is
rotated about the
y-axis.
 The result is a
cylindrical shell
with average
radius , height
, and thickness ∆x.
CYLINDRICAL SHELLS METHOD
( )
i
f x
i
x
( )
i
f x

14. Cylindrical Shells Method
Thus, by the formula the volume is
calculated as:
(2 )[ ( )]
i i i
V x f x x

 

15. Cylindrical Shells Method
So, an approximation to the volume
“V” of “S” is given by the sum of the
volumes of these shells:
1 1
2 ( )
n n
i i i
i i
V V x f x x

 
  
 

16. Cylindrical Shells Method
The approximation appears to
become better as n→∞. However,
from the definition of an integral,
We know that;
1
lim 2 ( ) 2 ( )
n b
i i a
n
i
x f x x x f x dx
 


 
 

17. Cylindrical Shells Method
 The volume of solid obtained by rotating
about the y-axis the region under the curve y
= f(x) from a to b is:
where 0 ≤ a < b
2 ( )
b
a
V xf x dx

 

18. Cylindrical Shells Method
Similarly, the volume
obtained by rotating about
the x- axis on [c, d] is:
V= 𝑐
𝑑
2𝜋𝑦𝑔 𝑦 𝑑𝑦

19. Cylindrical Shells Method(Example
1)
Find the volume of the solid obtained by
rotating about the y-axis the region
bounded by y = 2x2 - x3 and y = 0.
We see that typical
Shell has radius x,
and height
f(x) = 2x2 - 𝑥3.

20. Cylindrical Shell
Method(Example1)
So by Shell Method,
The volume is;
V = 0
2
(2𝜋𝑥)(2𝑥2
– 𝑥3
)dx
= 0
2
(2𝜋)(2𝑥3
‒ 𝑥4
)dx

21. Cylindrical Shells
Method(example2)
Find the volume of the solid obtained
by rotating about the y-axis the region
between y = x and y = x2.
We see that Shell
has radius x and
height x- 𝑥2
.

22. Cylindrical Shells
Method(example2)
 Thus, the volume of the solid is:
  
 
1
2
0
1
2 3
0
1
3 4
0
2
2
2
3 4 6
V x x x dx
x x dx
x x




 
 
 
  
 
 



23. Comparison Of Method