Lecture
XV
Iterated
Integrals
In
this
lecture
we
look at methods
to compute multiple integrals,
introducing
iterated integrals
.
First,
let us
define the type of
regions
for
which
it can
be used.
These
regions
are
called
simple
regions
.
Definition 1
A region
R
in
E
2
is
called
simple
if it
has
one
of the
following
properties:
(i) There
exists
an interval
[
a,
b
]
and there
exist
continuous
functions
g
1
(
x
)
and
g
2
(
x
)
defined on
[
a,
b
]
such that
g
1
(
x
)
≤
g
2
(
x
)
for
all
x
∈
[
a,
b
]
and
the
region
R
is the
set
of points
(
x,
y
)
such that
a
≤
x
≤
b
and
g
1
(
x
)
≤
y
≤
g
2
(
y
)
.
A region with this
property
is
called
y
simple
.
(ii) There
exists
an interval
[
c,
d
]
and there
exist
continuous
functions
h
1
(
y
)
and
h
2
(
y
)
defined on
[
c,
d
]
such that
h
1
(
y
)
≤
h
2
(
y
)
for
all
x
∈
[
c,
d
]
and
the
region
R
is the
set
of points
(
x,
y
)
such that
c
≤
y
≤
d
and
h
1
(
y
)
≤
x
≤
h
2
(
x
)
.
A region with this
property
is
called
x
simple
.
Definition 2
Let
R
be
an
y
simple
region in
E
2
, with
g
1
(
x
)
and
g
2
(
x
)
its
lower
and upper boundary functions, as
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 Two '04
 Duorg
 Math, Calculus, Derivative, Integrals, Multiple integral, g2, Iterated Integrals

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