Calculus 7.3 Day 2
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Transcript Calculus 7.3 Day 2
7.3 day 2
Disks, Washers and Shells
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:
5
Find the volume
the
1 dy
4region
1 4 y of
2
bounded by y x 1 , x 2 ,
5
and y 0
the y revolved
5 y dy 4about
1
axis.
5
4
3
y x2 1
2
5
1
5 y y 2 4
2 1
1
0
2
1
We can use the washer method ifwe split
25 itinto1 two
parts:
25 5 4
y 1 x2
5
2
2
1
outer
radius
x y 1
2
y 1 dy 2 1
inner
radius
2
cylinder
thickness
of slice
2
2
25 9
4
2 2
16
4
2
8 4
12
5
4
Here is another
way we could
approach this
problem:
3
y x2 1
2
1
0
1
2
cross section
If we take a vertical slice and revolve it about the y-axis
we get a cylinder.
If we add all of the cylinders together, we can reconstruct
the original object.
5
4
3
y x2 1
2
1
0
2
1
cross section
The volume of a thin, hollow cylinder is given by:
Lateral surface area of cylinder thickness
circumference height thickness
=2 r h thickness
=2 x x 2 1 dx
r
h
circumference thickness
r is the x value of the function.
h is the y value of the function.
thickness is dx.
5
4
This is called the
shell method
because we use
cylindrical shells.
3
y x2 1
2
1
0
2
1
cross section
If we add all the cylinders from the
smallest to the largest:
2
0
=2 r h thickness
=2 x x 2 1 dx
r
h
circumference thickness
2 x x 2 1 dx
2 4 2
2
2 x3 x dx
0
2
1 4 1 2
2 x x
2 0
4
12
Find the volume generated
when this shape is revolved
about the y axis.
4
3
2
1
0
1
2
3
y
4
5
6
4 2
x 10 x 16
9
7
8
We can’t solve for x,
so we can’t use a
horizontal slice
directly.
If we take a
vertical slice
and revolve it
about the y-axis
we get a cylinder.
4
3
2
1
0
Shell method:
1
2
3
y
4
5
6
4 2
x 10 x 16
9
7
8
Lateral surface area of cylinder
=circumference height
=2 r h
Volume of thin cylinder 2 r h dx
4
3
2
1
0
1
Volume of thin cylinder 2 r h dx
4 2
2 2 x 9 x 10x 16 dx
8
r
circumference
h
thickness
2
3
y
4
5
6
4 2
x 10 x 16
9
7
8
160
502.655 cm3
Note: When entering this into the calculator, be sure to enter
the multiplication symbol before the parenthesis.
When the strip is parallel to the axis of rotation, use the
shell method.
When the strip is perpendicular to the axis of rotation,
use the washer method.