Lecture 16  Wednesday May 6th
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16.1 Improper double integrals
An
improper integral
is an integral
R R
D
fdA
such that
D
is an unbounded region or
f
is
an unbounded function. For example, if
D
=
R
2
or
D
=
{
(
x,y
) :

x

y
 ≥
1
}
, then
D
is unbounded. Similarly, the integral of 1
/
(1

x
2

y
2
) over the unit disc
D
=
{
(
x,y
) :
x
2
+
y
2
≤
1
}
is also an improper integral. In this section, we shall only deal with improper
integrals of functions over rectangles or simple regions
D
, such that
f
has only ﬁnitely
many discontinuities in
D
.
We recall that if
f
:
R
→
R
is continuous at all points in [
a,b
] except at some point
z
∈
[
a,b
], then we may determine the integral of
f
on [
a,b
] using limits:
Z
b
a
f
(
x
)
dx
= lim
c
→
z

Z
c
a
f
(
x
)
dx
+ lim
c
→
z
+
Z
b
c
f
(
x
)
dx
when each of the integrals on the right exists. In other words, we take a ball around
z
and then let the radius of the ball shrink to zero in the limit. This deals with the single
integral of a function which may be unbounded at a point
z
, and we can extend this
to cases where the function has ﬁnitely many discontinuities. For double integrals, we
proceed in a similar way. Suppose
z
is a discontinuity of a function
f
that is continuous
at every other point of a bounded rectangle or simple region
D
. Let
B
z
be an open ball
of radius
δ
containing
z
. Then
f
is continuous on
D
(
δ
) =
D
\
B
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 Spring '07
 Enright
 Math, Calculus, Improper Integrals, Integrals, lim, 1 2 2 g, [email protected]

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