6.3 – Volumes of Cylindrical
Shells
Derivation
Assume you have a functions similar to the one shown
below and assume the f is to difficult to solve for y in
terms of x. Rotate f about the y axis.
y = f (x)
xi
xo
It is too difficult to find xi
and xo so that we can find
the area of the washer. We
need another method for
more complicated
functions.
Cylindrical Shells
Let xi be some subinterval and establish a rectangle to
estimate the area. When this rectangle is rotated about the
y-axis, a cylindrical shell if formed.
y = f (x)
xi
r1
Δr
r2
h
Cylindrical Shells
Now determine the volume of the cylindrical shell. V2 is the volume
of the larger right circular cylinder and V1 is the volume of the inner.
V  V2  V1
  r22 h   r12 h

  r

 r r  r h

r r 
r  r h
 2
 r r h
2
2
2
1
2
1
2
2
1
1
2
V  2r  h  r
2
r1
Δr
1
V  [circum ference][height][thickness]
r2
h
V  [circumference][height][thickness]
Cylindrical Shells
Now assume that the interval [a, b] is divided into n
subintervals of equal width Δx. Also assume that xi is taken
at the mid-point of each subinterval. This means that ri = xi
the height of each cylindrical shell is given by f (xi). The
volume of each cylindrical shell is
Vi  (2 xi ) f ( xi )x
Cylindrical Shells
An estimate of the total volume is the sum of the
volumes of the n cylindrical shells.
n
V   (2 xi ) f ( xi )x
i 1
Volume of a Solid Using Cylindrical Shells
Therefore, the volume of the solid obtained by rotating
about the y-axis the region under the curve y = f (x) from
a to b is
n
b
i 1
a
V  lim  2 xi f ( xi ) x   2 x f ( x) dx
n 