MATH 232 Lecture 6: 8.9 Improper Integrals
Impwputntegrals
you
have
covered
definite
integrals
Sab
f
Ix
)
dx
with
f
defined
on
a
finite
interval
[
a
,
b
]
with
no
infinite
discontinuities
.
(
in
calculus
1)
In
this
section
we
will
cover
the
case
where
the
interval
is
infinite
and
also
the
case
where
f-
has
an
infinite
discontinuity
in
La
,
b
]
.
I
×
÷"÷÷x
interval
is
infinite
infinite
dais
continuity
In
either
case
the
integral
is
called
an
improperintegral
.
Type1::InfiniteInterval
Consider
the
infniteregion
ER
B-
that
lies
under
the
curve
y
-
-
La
,
above
the
X
-
axis
and
to
the
night
of
X
-
-
I
.
We
will
show
that
the
areaoythisfnite
.
The
area
of
the
part
n
R
to
the
left
a
X
-
-
t
is
A
Lt
)
=
f
!
¥
d
x
-
-
-
I
=
I
-
I
Note
that
A
Ct
)
s
I
no
matter
how
large
t
is
chosen
.
Note
that
thing
.
A
ft
)
-
-
thin
,
(
I
-
I
)
=
I
R
it
Styrofoams
Therefore
,
the
area
of
the
shaded
region
approaches
l
as
t
→
a
.
We
say
that
the
areaoltheinfiniteregionkis
equal
to
1
and
we
write
:
I
?
¥
dx
-
-
fusing
f
!
¥
d
x
-
-
I
¥dx=t÷xdx÷I
ii.
t
:÷÷:÷:¥÷÷¥÷÷÷÷÷÷÷÷
axis
?
#
dx
-
-
fun's
.
Six
-
sdx
-
-
Engine
it
!
)
-
-
fins
.
f-
II.
+
E)
-
-
I
infinite
region
,
finite
area
T
'
⑧
\
Y
=
Tz
area
-
-
I
÷
i
:*
:
.
"
area
=
If
#zdx=÷lxi)¥GD
infinite
region
,
=D
infinite
area
!
-
is
y
=
I
infect
area
"
y
=
1×-2
finite
area
i
yji¥
.
⇐
nite
area
'
1256
>
x
finite
area
The
last
problem
suggests
that
finite
area
depends
$
on
how
'
'
fast
"
the
canoe
dives
toward
the
x
-
axis
.
We
will
return
to
this
idea
after
we
formalize
the
definition
of
a
type
1
improper
integral
.
Document Summary
Impwputntegrals you have covered definite integrals sab fix) dx with f defined on a finite interval [a with no infinite discontinuities . In this section we will cover the case where the interval an infinite discontinuity in la , is infinite and also the case where f- has b] . ( in calculus 1) In either case the integral is called an improperintegral . X- axis and to the night of x- The area of the part n r to the left a x - above the. Note that a ct) s i is chosen . no matter how large t. Therefore , the area of the shaded region approaches l as t a . areaoltheinfiniteregionkis. We say that the equal to 1 and we write : Engine it!) infinite region , finite area t" "1256 y = i y = 1 -2 yji . > x infect area finite area (cid:12200)nite area finite area.
