7.3 day 3
The Shell Method
Grows
to over
12 feet
wide
Japanese
Spider
Crab
and lives
100 years.
Georgia
Aquarium,
Atlanta
Photo by Vickie Kelly, 2006
Greg Kelly, Hanford High School, Richland, Washington
5
Find the volume
the 4region
1 dy

1 4 y yof
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 ifwe split
25  itinto1 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.
