Math 104 – Rimmer
6.3 Volumes by Cylindrical Shells
Sometimes finding the volume of a solid of revolution is
impossible by the disk or washer method
Since there is a gap b/w the region and the
axis of rotation, we would try ____________
( )
y = sin x 2
We would have to solve for __ as a function of __
since the axis of rotation is ________.
π
Sometimes this is the problem, but we can do it here.
x=
Our problem is that the outer radius and the inner
radius use the _________ .
In order to find the volume of this solid of revolution we need a
different technique.
Math 104 – Rimmer
6.3 Volumes by Cylindrical Shells
The ___________________ uses the volume of _______
__________ to find the volume of a solid of revolution.
To understand the formula, lets first look at one of the cylindrical
shells:
There are two cylinders, an outer
r
router
rinner
cylinder and an inner cylinder.
The volume of the “shell” we use is found by
taking the volume of the inner cylinder and
______________the volume of the outer cylinder.
Vshell =
Vshell =
h
Vshell =
Vshell =
Vshell =
( router + rinner ) = r
2
Let r = raverage
average
Let ∆r = router − rinner
Vshell =
1
Math 104 – Rimmer
6.3 Volumes by Cylindrical Shells
( )
y = sin x
2
Vshell =
2π r
⋅ h ⋅ ∆r
circumference height thickness
Vi =
now ______ the volume
of all the shells
π
V=
you get a __________ as the number of shells ____
n
V = lim ∑ 2π xi ⋅ f xi ⋅ ∆x
n →∞
( ) ( )
i =1
V=
Volume of the solid obtained
( )
f xi
by rotating the region under
the curve f ( x ) from x = a to x = b
xi
about the y − axis
xi −1
xi
xi
Math 104 – Rimmer
6.3 Volumes by Cylindrical Shells
In general,
b
V = ∫ 2π ( radius )( height ) dx
a
Typical
Vertical axis
Horizontal axis
rectangle
of rotation
of rotation
Disk or
Washer
Cylindrical
Shells
2
Math 104 – Rimmer
6.3 Volumes by Cylindrical Shells
( )
y = sin x
2
radius =
height =
b
π
V = ∫ 2π ( radius )( height ) dx
a
V=
=
=
=
=
Set up, but do not evaluate, an integral for the volume
obtained by rotating the region bounded by
1
π
y = cos 2 x, y = , about the line x =
4
2
Math 104 – Rimmer
6.3 Volumes by Cylindrical Shells
1


2
 below y = cos x and above y = , from − a to a where these are the intersection pts. closest to the y -axis 
4


radius =
height =
limits of integration ⇒
b
V = ∫ 2π ( radius )( height ) dx
a
3