Calculus 7.3 Day 2 - University of Houston
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Transcript Calculus 7.3 Day 2 - University of Houston
7.3 day 2
Disk and Washer Methods
Limerick Nuclear Generating Station, Pottstown, Pennsylvania
Photo by Vickie Kelly, 2003
Greg Kelly, Hanford High School, Richland, Washington
2
Suppose I start with this curve.
y x
My boss at the ACME Rocket
Company has assigned me to
build a nose cone in this shape.
1
0
1
2
3
4
So I put a piece of wood in a
lathe and turn it to a shape to
match the curve.
2
How could we find the volume
of the cone?
y 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 2 the thickness
x
2
dx
In this case:
r= the y value of the function
thickness = a small change
in x = dx
2
The volume of each flat
cylinder (disk) is:
y x
r 2 the thickness
1
0
1
2
3
x
2
dx
4
If we add the volumes, we get:
x
2
4
0
dx
4
x dx
0
2
4
8
x2
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:
b
V y 2 dx
a
Since we will be using the disk method to rotate shapes
about other lines besides the x-axis, we will not have this
formula on the formula quizzes.
b
A shape rotated about the y-axis would be:
V x 2 dy
a
1
The region between the curve x
, 1 y 4 and the
y
y-axis is revolved about the y-axis. Find the volume.
y
x
1
1
2
3
4
1
.707
2
1
.577
3
1
2
We use a horizontal disk.
The thickness is dy.
4
3
2
The radius is the x value of the
1
function
.
dy
y
1
2
1
V
dy
y
1
4
0
1
4
1
1
dy
y
volume of disk
ln y 1
4
0
ln 4 ln1
ln 22 2 ln 2
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:
500 ft
x
x .000574 y 2 .439 y 185
The volume can be calculated using the disk method with
a horizontal disk.
500
0
.000574 y
2
.439 y 185 dy 24,700,000 ft 3
2
4
3
y 2x
2
y x2
The region bounded by
y x2 and y 2 x is
revolved about the y-axis.
Find the volume.
1
If we use a horizontal slice:
yx
y 2x
y
x
2
2
yx
0
1
2
The volume of the washer is:
V
0
4
y
2
y
2
2
dy
1 2
V y y dy
0
4
4
V
4
0
1 2
y y dy
4
The “disk” now has a hole in
it, making it a “washer”.
R
R
2
r 2 thickness
2
r 2 dy
outer
radius
4
1
1
y 2 y3
12 0
2
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:
b
V R2 r 2 dx
a
Like the disk method, this formula will not be on the
formula quizzes. I want you to understand the formula.
y x2
4
3
y 2x
2
1
0
1
r
y 2x
y
x
2
y x2
yx
2
r 2 y
y2
4 2 y 4 4 y y dy
0
4
1
4
1 2
3 y y 4 y 2 dy
0
4
4
V R 2 r 2 dy
0
2
y
2 2 y
0
2
dy
2
4
3
1
8
y 2 y3 y
12
3 0
2
16 64
8
24
3
3 3
3
2
y2
4 2 y 4 4 y y dy
0
4
4
The outer radius is:
y
R 2
2
The inner radius is:
R
4
4
If the same region is
rotated about the line x=2: