15.1: DOUBLE INTEGRALS OVER RECTANGLES
KIAM HEONG KWA
1.
Volumes and Double Integrals
Let
f
be a nonnegative continuous function of two variables deﬁned
on a closed rectangle
R
= [
a,b
]
×
[
c,d
] =
{
(
x,y
)
∈
R
2

a
≤
x
≤
b,c
≤
y
≤
d
}
,
and let
S
=
{
(
x,y,z
)
∈
R
3

0
≤
z
≤
f
(
x,y
)
,
(
x,y
)
∈
R
}
be the solid that lies above
R
and under the graph of
f
. Before giving
a precise meaning to the volume
V
=
V
(
S
) of
S
, we ﬁrst do an approx
imate calculation of how much space the solid
S
occupies. Of course, a
very very very
crude approximation can be done by simply choosing at
random a sample point (
x
*
,y
*
) in
R
and then evaluating the product
of
f
(
x
*
,y
*
) and the area of
R
:
V
≈
f
(
x
*
,y
*
)
×
the area of
R.
However, this is usually a
very very very
bad approximation, unless
f
varies very little over the rectangle
R
.
Intuitively, it is clear that the variation (in values) of a continuous
function decreases as we decrease the size of the region over which the
variation is measured. Hence we shall ﬁrst divide the solid
S
into solids
S
ij
of much smaller dimensions and approximate the volume
V
ij
of each
S
ij
. The precise deﬁnition for
S
ij
will be given in the sequel.
We divide the base of
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 Fall '10
 Kwa
 Calculus, Geometry, Integrals, Angles, rij, sij, KIAM HEONG KWA, )∆A,

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