Section 7.3 – Volume: Shell
Method
Calculate the volume of the solid obtained by
rotating the region bounded by y = x2, x=0, and y=4
about x = -1.

4
0




Calculator
White Board Challenge
56

y  1   0  1  
 3
2
2
We will now investigate another
method to calculate this volume.
Volume of a Shell
Consider the following cylindrical shell (formerly a washer):
Imagine the circle in
in the middle of the
base area.
C  2 R
rinner
router
R
router  rinner
R
2
The average of the
radii is a new radius
from the center of
the base to the
middle of the
enclosed area.
h
Thus, the
circumference of the
middle circle is…
Label the new
radius.
Δr
Also, the thickness of
the shell is…
r  router  rinner
Volume of a Shell
The volume of the cylindrical shell is easier to see
The cylindrical shell
when it is flattened out:
The length of the
base is…
C = 2πR
flattened out is a
rectangular prism.
The height of the
base is…
h
Thus the volume
of the prism is…
V  2 R  h  r
The height
of the
prism is…
Δr
Volumes of Solids of Revolution with
Riemann Sums
Let us rotate the region under y=f(x) from x=a to x=b about the yaxis. The resulting solid can be divided into thin concentric shells.
thickness
Volume 

Height
lim
max xk 0
n

2 R  h  t
k 1
b
  2 R  h  dx
a
b
radius
a
b
 2  R  h  dx
a
Volumes of Solids of Revolution:
Shell Method
MAKE A HOOK:
• Sketch the bounded region and the line of Opposite of
Washer
revolution.
Method
• If the line of revolution is horizontal, make sure the
equations can easily be written in the x= form. If
vertical, the equations must be in y= form.
• Sketch a generic shell (a typical cross section).
• Find the radius of the generic shell (perpendicular
distance from the line revolution to the outer edge of
the shell), the height of the shell (this length is
perpendicular to the radius), and the thickness of
the shell (this length is perpendicular to the height).
• Integrate with the following formula:
b
V  2  radius  height  thickness
a
Example 1
Calculate the volume of the solid obtained by rotating the
region bounded by y = x2, x=0, and y=4 about x = -1.
Sketch a Graph
Find the Boundaries/Intersections
Radius
Thickness
= x – -1 = x + 1
= dx
Height
= 4 – x2
Make Generic
Shell(s)
Line of Rotation
x0
x2  4
x  x 2
We only
need
x>2
Integrate the Volume of the
Shell
2   x  1   4  x  dx
2
2
0
56
 
3
Example 2
Calculate the volume V of the solid obtained by rotating the region
bounded by y = 5x – x2 and y = 8 – 5x + x2 about the line y-axis.
Sketch a Graph
Find the Boundaries/Intersections
Thickness
2
2
5x  x  8  5x  x
= dx
Line of Rotation
x  1, 4
Integrate the Volume of Each
Generic Washer
2  x   5 x  x 2    8  5 x  x 2   dx
1
4
Radius = x
Height =
( 5x – x2 ) – (8 – 5x + x2)
Make Generic Shell(s)
 45
Use the shell method to calculate the volume of
the solid obtained by rotating the region bounded
by y = x1/2 and y=0 over [0,4] about the x-axis.
Calculator
White Board Challenge
Line of Rotation
Height = 4 – y2
Thickness
= dy
2  y   4  y
2
Radius = y
0
 8
2
 dy