Math 2224 Multivariable Calculus – Sec. 13.2: Double Integrals over General Regions
I.
Types of Regions
We will be discussing two types of general regions in this section
A.
Type I
An
xy
region on the
xy
plane is called a type I region if any vertical strip (in the
y
direction) always has the same upper and lower boundaries and the set can be described
by the set of inequalities
a
!
x
!
b
,
g
1
(
x
)
!
y
!
g
2
(
x
)
.
B.
Type II
An
xy
region on the
xy
plane is called a type II region if any horizontal strip (in the
x
direction) always has the same right and left boundaries and the set can be described by
the set of inequalities
c
!
y
!
d
, h
1
(
y
)
!
x
!
h
2
(
y
)
.
C.
Type I
Type II
both types
II.
Fubini’s Theorem (Stronger Form)
Let
f(x,y)
be continuous on a region
R.
1. If
R
is defined by
a
!
x
!
b
,
g
1
(
x
)
!
y
!
g
2
(
x
),
with
g
1
(
x
)
and
g
2
(
x
)
continuous on
[
a,b
], then
f
(
x
,
y
)
dA
R
!!
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 Spring '03
 MECothren
 Multivariable Calculus, xy plane, Fubini, gio ns I., Lower limit=bottom curve

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