Volumes
using Disks
Limerick Nuclear Generating Station, Pottstown, Pennsylvania
Photo by Vickie Kelly, 2003
Greg Kelly, Hanford High School, Richland, Washington
2
y
Suppose I start with this curve.
x
And we want to make a cone
using this as the shape.
1
0
1
2
3
4
2
y
How could we find the volume
of the cone?
x
One way would be to cut it into a
series of thin slices (flat cylinders)
and add their volumes.
1
0
1
2
3
4
The volume of each flat
cylinder (disk) is:
r the thickness
2
x
2
dx
In this case:
r= the y value of the function
thickness = a small change
in x = dx
2
y
The volume of each flat
cylinder (disk) is:
x
r the thickness
1
0
2
1
2
3
The Rectangle is
Perpendicular
to the Axis
4
x
2
dx
If we add the volumes, we get:
4
x
0
2
4
2
dx
x dx
0
4
x
8
2
0
This application of the method of slicing is called the
disk method. The shape of the slice is a disk, so we
use the formula for the area of a circle to find the
volume of the disk.
If the shape is rotated about the x-axis, then the formula is:
Volume
b
f ( x ) dx
2
a
The Rectangle is Perpendicular to the X-Axis
If the Rectangle is
Perpendicular to the Y-Axis
V
b
f ( y ) dy
2
a
The region between the curve x
1
y
,
1 y 4
and the
y-axis is revolved about the y-axis. Find the volume.
y
x
1
1
2
3
4
1
3
.707
2
2
1
3
We use a horizontal disk.
The thickness is dy.
4
.577
The radius is the x value of the
1
function
.
dy
y
1
1
2
0
1
V
4
1
1
y
2
dy
4
1
1
dy
y
volume of disk
0
ln y 1 ln 4 ln 1
4
ln 4
Find the volume of rotating the area
enclosed by the three restrictions about:
the x-axis and x = 2
y x
y0
x2
2
Answers:
x-axis
32
5
x=2
8
3
Food items that have a disk-based volume.
Hamburgers
French Bread
Cookies
Volumes using Shells
(Washers)
Limerick Nuclear Generating Station, Pottstown, Pennsylvania
Photo by Vickie Kelly, 2003
Greg Kelly, Hanford High School, Richland, Washington
y
The natural draft cooling tower
shown at left is about 500 feet
high and its shape can be
approximated by the graph of
this equation revolved about
the y-axis:
5 0 0 ft
x .000574 y .439 y 185
2
x
The volume can be calculated using the disk method with
a horizontal disk.
.000574 y
500
0
2
.439 y 185
2
dy 24, 700, 000 ft 3
4
3
y 2x
2
y x
2
The region bounded by
2
y x and y 2 x is
revolved about the y-axis.
Find the volume.
1
If we use a horizontal slice:
y x
y 2x
2
y
y x
0
1
2
x
2
The volume of the washer is:
V
4
0
V
4
0
V
y
2
2
y
dy
2
1 2
y y dy
4
4
0
y
1
4
The “disk” now has a hole in
it, making it a “washer”.
R
2
r
R r
2
outer
radius
4
2
y dy
1 3
1 2
y
y
12
2
0
2
2
thickness
dy
inner
radius
16
8
3
8
3
This application of the method of slicing is called the
washer method. The shape of the slice is a circle
with a hole in it, so we subtract the area of the inner
circle from the area of the outer circle.
The washer method formula is:
V
b
R r
2
2
dx
a
1) Draw the graphs
2) Draw two rectangles perpendicular to the axis
3) Integrate Outside - Inside
y x
4
3
2
y 2x
If the same region is
rotated about the line x=2:
The outer radius is:
2
R 2
1
0
1
2
2
The inner radius is:
r
y x
y 2x
2
y
y x
R
r 2
x
2
V
4
R r dy
2
4
0
0
4
4 2y
y
y
2
44
0
y
2 2
2
2
y
4 2y
44
4
y
2
dy
y y dy
4
0
2
2
4
y
4
0
3 y
1
1
y 4 y 2 dy
2
4
y y dy
4
3 2
1 3 8 2
y
y y
12
3
2
0
3
16 64
24
3
3
8
3
5
5
4
3
y x 1
2
2
Find the volume
1 dthe
y 4region
1 4 y of
2
bounded by y x 1 , x 2 ,
5
and y 0 revolved
the y5 y d y 4about
1
axis.
5
1 2
5 y y 4
2
1
1
0
2
1
We can use the washer method if we split
25 it into1 two
parts:
y 1 x
5
2
2
1
outer
radius
2
x
y 1
inner
radius
2
25
y 1
dy 2 1
2
5 4
2
2
25 9
4
2
2
cylinder
thickness
of slice
16
4
2
8 4
1 2
0
