7.3 Day One: Volumes by Slicing
x2
x2
x
2
2
8
15 2
x
4
3 2
4 x
Find the volume of the pyramid:
Consider a horizontal slice through
the pyramid.
3
The volume of the slice is s2dh.
If we put zero at the top of the
pyramid and make down the
positive direction, then s=h.
3
3
Vslice h 2 dh
0
h
s
3
3
1 3
V h dh h 9
0
3 0
3
dh
2
This correlates with the formula:
1
1
V Bh 9 3 9
3
3
Method of Slicing:
1
Sketch the solid and a typical cross section.
2
Find a formula for V(x).
(Note that I used V(x) instead of A(x).)
3
Find the limits of integration.
4
Integrate V(x) to find volume.
A 45o wedge is cut from a cylinder of radius 3 as shown.
Find the volume of the wedge.
You could slice this
wedge shape several
x ways, but the simplest
cross section is a
rectangle.
y
If we let h equal the height of the slice then the volume of
the slice is: V x 2 y h dx
Since the wedge is cut at a
Since
x y 9
2
2
h
45o
angle:
45o
x
hx
y 9 x2
y
x
Even though we
started with a
cylinder, does not
enter the
calculation!
V x 2 9 x2 x dx
2
V
x
2
y
h
dx
u
9
x
u 0 9
3
2
V 2 x 9 x dx du 2x dx u 3 0 h x
0
0
1
2
V u du
9
2
u
3
3 9
2
2
y 9 27
x 2 18
3
0
Cavalieri’s Theorem:
Two solids with equal altitudes and identical parallel cross
sections have the same volume.
Identical Cross Sections
Cavalieri’s Theorem: Volume of a Sphere
V=
2
=
2
sin x
2
dx
0
1
0 2 1 cos 2x dx
sin 2x
=
x+
4
2 0
=
4
2
=
4
7.3 Disk and Washer Methods
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
b
A shape rotated about the y-axis would be:
V x dy
2
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
4
0
1
1
1
dy
y
volume of disk
0
2
ln y 1 ln 4 ln1 ln 2 2 ln 2
4
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 x 2 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
V y y 2 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 thickness
R r dy
2
2
2
2
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 R 2 r 2 dx
a
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 R2 r 2 dy
0
2
y
2 2 y
0
2
dy
2
4
3 2 1 3 8
y y 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:
Washer Cross Section
The region in the first quadrant enclosed by the y-axis
and the graphs of y = cos x and y = sin x is revolved
about the x-axis to form a solid. Find its volume.
Washer Cross Section
The region in the first quadrant enclosed by the y-axis
and the graphs of y = cos x and y = sin x is revolved
about the x-axis to form a solid. Find its volume.
V=
4
0
=
A(x) dx =
0
4
0
4
cos x - sin x dx
cos 2x dx
sin 2x 4
3
=
=
units
2
2 0
2
2
7.3 The Shell Method
5
Find the volume
the 4region
1 dy
1 4 y yof
2
x 1, x 2 ,
bounded by
5
and y 0 revolved
the y5 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
Japanese Spider Crab of slice
Georgia Aquarium, Atlanta
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
1
2
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
1
2
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 10 x 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.
Find the volume of the solid when the region bounded by the
curve y = x , the x-axis, and the line x = 4 is revolved about
the x-axis. Find the volume of the solid using cylindrical
shells.
Find the volume of the solid when the region bounded by the
curve y = x , the x-axis, and the line x = 4 is revolved about
the x-axis. Find the volume of the solid using cylindrical
shells.
V=
2
0
2 y 4 - y
= 8
Radius = y
x = y2
Shell height = 4 - y 2
2
dy
Find the volume of the solid of revolution formed
by revolving
2
the region bounded by the graph of x e y and the y
axis, 0 ≤ y ≤ 1, about the x-axis. Use the Shell Method.
2
1
2
-1
Find the volume of the solid of revolution formed
by revolving
2
the region bounded by the graph of x e y and the y
axis, 0 ≤ y ≤ 1, about the x-axis. Use the Shell Method.
1
V 2 ye
y2
0
1.986
2
1
e y
0
1
1
e
2
dy
1
2
-1
Find the volume of the solid formed by revolving the region
bounded by the graphs y = x3 + x + 1, y = 1, and x = 1 about
the line x = 2.
3
axis of
revolution
2
1
2
-1
4
Find the volume of the solid formed by revolving the region
bounded by the graphs y = x3 + x + 1, y = 1, and x = 1 about
the line x = 2.
1
0
3
axis of
revolution
2
2 x 4 2x 3 x 2 2x dx
0
2
1
2
4
1
x
x
x
2
=2
x
4
3
5
0
1 1 1
2 1
5 2 3
5
-1
V 2 2 x x 3 x 1 1 dx
29
15
4
3
0
