Tuesday, November 22
−
Lecture 30 :
Volumes of a solid of revolution: Cross
section (slicing) method
(Refers to Section 7.2 in your text)
Students who have mastered the content of this lecture know
:
what computing the volume of solid by
crosssection means, about solids of revolution, the formula for computing the volume of a solid of
revolution by crosssections.
Students who have practiced the techniques presented in this lecture will be able to
:
Compute the
volume of solids by using crosssections in cases where the solid is a solid of revolution and cases where
it is not.
30.1
Introduction
−
In this lecture we study methods for determining the volume of a few
simple solids for which we can obtain a formula for its crosssections.
30.1.1
Definition
−
Calculating the volume by the
method of cross sections
(referred
to as
slicing
in the text) is done by determining the area
A
(
x
) of vertical cross sections
of a solid at
x
, or the area
A
(
y
) of a horizontal cross section of the solid at
y
.
30.2
The principle behind the cross section method
.

Suppose we are given a solid
S
lying alongside, "
skewed
" by the
x
axis, whose
extremities are given by the planes
x = a
and
y = b
, in 3space.

Let us subdivide the interval [
a, b
] into
n
equal subintervals
a
=
x
0
,
x
1
,
x
2
,
...,
x
n
−
1
,
x
n
= b,
each of length
∆
x =
(
b
−
a
)
/ n
.

Let
A
(
x
i
) denote the area of a
cross section
of the solid over the interval [
x
i
−
1
,
x
i
] i.e.,
the area of the region formed by the intersection of the plane
x
=
x
i
in 3space with the
solid
S
. The symbol
x
i
represents the right endpoint of the
i
th
interval
∆
x
.

Thus
A
(
x
1
),
A
(
x
2
),
...,
A
(
x
n
) represent a sequence of areas of cross sections of the
solid
S
at
x
1
,
x
2
,
...,
x
n
−
1
,
x
n
.