The document discusses various methods for calculating the volume of solids of revolution in calculus, specifically focusing on the area between curves, disk, washer, and cylindrical shell methods. It provides detailed explanations and examples for each approach, highlighting the steps needed for integration based on vertical and horizontal strips. The content is aimed at assisting students in the chemical engineering branch during their first semester of the academic year 2014.

1. Active Learning Assignment
CALCULAS(2110014)
Branch : Chemical Engineering
Sem : 1st
Academic Year : 2014(odd)

2. APPLICATION OF INTEGRATION

3. Objectives
 Find the volume of a solid of revolution using
the area between the curves method.
 Find the volume of a solid of revolution using the volume slicing
method.
 Find the volume of a solid of revolution using
the disk method.
 Find the volume of a solid of revolution using the washer method.
 Find the volume of a solid of revolution using the cylindrical shell
method.

4. Areas Between Curves
To find the area:
• divide the area into n strips of
equal width
• approximate the ith strip by a
rectangle with base Δx and height
f(xi) – g(xi).
• the sum of the rectangle areas is a
good approximation
• the approximation is getting better
as n→∞.
y = f(x)
y = g(x)
• The area A of the region bounded by the curves y=f(x), y=g(x),
and the lines x=a, x=b, where f and g are continuous and f(x) ≥
g(x) for all x in [a,b], is
b
A [ f (x) g(x)]dx
  
a

5. 2
y1  2  x
2y  x
2
2
1
2 x x dx

  
2
1 1
3 2
1
2
x x x
3 2

 
8 1 1
   
 4   2   2
  
 3   3 2

8 1 1
6   2
 
3 3 2
    27
36 16 12 2 3
6
6

9
2

Example:-find area of the region enclosed by parabola
2
and the line . And .
1 2yx 2yx
solution:

6. Example: Find the area of the region by parabola and the line .
y  x
y  x  2
yx
y  x  2
If we try vertical strips, we
have to integrate in two parts:
dx
dx
2 4
 
0 2
 x dx   x  x  2 dx
We can find the same area
using a horizontal strip.
dy
Since the width of the strip
is dy, we find the length of
the strip by solving for x in
y  x terms of y.
2 y  x
y  x  2
y  2  x
 
2
2
0
 y  2  y dy
2
1 1
 
2 3
0
y  2
y  y
2 3
8
2 4
3
10
3

dx
yx 2yx
solution:

7. General Strategy for Area Between Curves:
1
Sketch the curves.
Decide on vertical or horizontal strips. (Pick
whichever is easier to write formulas for the length of
the strip, and/or whichever will let you integrate fewer
times.)
2
3 Write an expression for the area of the strip.
(If the width is dx, the length must be in terms of x.
If the width is dy, the length must be in terms of y.
4 Find the limits of integration. (If using dx, the limits
are x values; if using dy, the limits are y values.)
5 Integrate to find area.

8. Volume by slicing

14. The Disk Method

15. The Disk Method
If a region in the plane is revolved about a line, the
resulting solid is a solid of revolution, and the line is
called the axis of revolution.
The simplest such solid is a right
circular cylinder or disk, which is
formed by revolving a rectangle
about an axis adjacent to one
side of the rectangle,
as shown in Figure 7.13.
Axis of revolution
Volume of a disk: R2w
Figure 7.13

16. The Disk Method
The volume of such a disk is
Volume of disk = (area of disk)(width of disk)
= πR2w
where R is the radius of the disk and w is the
width.

17. The Disk Method
To see how to use the volume of a disk to find the
volume of a general solid of revolution, consider a solid
of revolution formed by revolving the plane region in
Figure 7.14 about the indicated axis.
Disk method
Figure 7.14

18. The Disk Method
To determine the volume of this solid, consider a
representative rectangle in the plane region. When
this rectangle is revolved about the axis of
revolution, it generates a representative disk whose
volume is
Approximating the volume of the solid by n such
disks of width Δx and radius R(xi) produces
Volume of solid ≈

19. The Disk Method
This approximation appears to become better
and better as So, you can define
the volume of the solid as
Volume of solid =
Schematically, the disk method looks like this.

20. The Disk Method
A similar formula can be derived if the axis of revolution is
vertical.
Horizontal axis of revolution Vertical axis of revolution
Figure 7.15

21. Example 1 – Using the Disk Method
Find the volume of the solid formed by revolving the
region bounded by the graph of and the
x-axis (0 ≤ x ≤ π) about the x-axis.
Solution:
From the representative
rectangle in the upper graph
in Figure 7.16, you can see that
the radius of this solid is
R(x) = f (x)
Figure 7.16

22. Example 1 – Solution
So, the volume of the solid of revolution is
Apply disk method.
Simplify.
Integrate.
[cont’d]

23. The Washer Method

24. The Washer Method
• The disk method can be extended to cover solids of revolution
with holes by replacing the representative disk with a
representative washer.
• The washer is formed by revolving a rectangle about an
axis,as shown in Figure 7.18.
• If r and R are the inner and outer radii
of the washer and w is the width of the
washer, the volume is given by
• Volume of washer = π(R2 – r2)w.
Axis of revolution Solid of revolution
Figure 7.18

25. The Washer Method
To see how this concept can be used to find the volume
of a solid of revolution, consider a region bounded by
an outer radius R(x) and an inner radius r(x), as shown
in Figure 7.19.
Figure 7.19
Plane region
Solid of revolution
with hole

26. The Washer Method
• If the region is revolved about its axis of
revolution, the
volume of the resulting solid is given by
Washer method
• Note that the integral involving the inner radius
represents
• the volume of the hole and is subtracted from the
integral involving the outer radius.

27. Example 3 – Using the Washer Method
Find the volume of the solid formed by revolving the region
bounded by the graphs of about the
x-axis, as shown in Figure 7.20.
Figure 7.20
Solid of revolution

28. Example 3 – Solution
In Figure 7.20, you can see that the outer and
inner radii are as follows.
Outer radius
Inner radius
Integrating between 0 and 1 produces
Apply washer method.

29. Example 3 – Solution
Simplify.
Integrate.
cont’d

30. The Washer Method
So far, the axis of revolution has been horizontal
and you have integrated with respect to x. In the
Example 4, the axis of revolution is vertical and
you integrate with respect to y. In this example,
you need two separate integrals to compute the
volume.

31. Example 4 – Integrating with Respect to y, Two-
Integral Case
Find the volume of the solid formed by revolving the region
bounded by the graphs of y = x2 + 1, y = 0, x = 0, and x = 1 about
y-axis, as shown in Figure 7.21.
Figure 7.21

32. Example 4 – Solution
• For the region shown in Figure 7.21, the outer
radius is
simply R = 1.
• There is, however, no convenient formula that
represents the inner radius
• When 0 ≤ y ≤ 1, r = 0, but when 1 ≤ y ≤ 2, r is
determined by the equation y = x2 + 1, which
implies that

33. Example 4 – Solution
Using this definition of the inner radius, you can
use two integrals to find the volume.
Apply washer method.
Simplify.
Integrate.
cont’d

34. Example 4 – Solution
• Note that the first integral represents the
volume
• of a right circular cylinder of radius 1 and height 1.
• This portion of the volume could have been
determined
• without using calculus.
cont’d

35. VOLUME BY CYLINDRICAL SHELL

41. Conclusion
We Hope this ppt usefull for you to find area and
volume with use of different methods likes disk ,
washer and cylindrical.

42. THANKS……….
Reference:
google