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6.3: Volumes by Cylindrical Shells
(Dated: October 5, 2011)
Consider a cylindrical shell with inner radius
r
1
, outer ra
dius
r
2
, and height
h
. Its volume is
V
=
π
(
r
2
2

r
2
1
)
h
=
π
(
r
1
+
r
2
)(
r
2

r
1
)
h
=
2
π
‡
r
1
+
r
2
2
·
h
(
r
2

r
1
)
=
2
π
rh
Δ
r
,
where
r
=
r
1
+
r
2
2
is the
average
radius and
Δ
r
=
r
2

r
1
is the
thinkness of the shell. It follows that one has the volume of
the cylindrical shell given by
V
=
circumference
×
height
×
thickness.
Let
S
be the solid obtained by rotating about the
y
axis
the region bounded b the curve
y
=
f
(
x
)
≥
0,
y
=
0,
x
=
a
,
and
x
=
b
, where 0
<
a
≤
b
. The volume of
S
can be ap
proximated by a Riemann sum whose summands are the
volumes of constituent cylindrical shells. To see this, let’s
divide the interval [
a
,
b
] into
n
subintervals of equal width
Δ
x
=
b

a
n
. Let [
x
i

1
,
x
i
] be the
i
th subinterval and let
x
i
be the midpoint of [
x
i

1
,
x
i
]. Note that if the rectangle
with base [
x
i

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This note was uploaded on 11/11/2011 for the course MATH 152.01 taught by Professor Geline during the Fall '09 term at Ohio State.
 Fall '09
 GELINE
 Calculus, Geometry

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