Double Integrals: General Region II
John E. Gilbert, Heather Van Ligten, and Benni Goetz
In evaluating a double integral
D
f
(
x, y
)
dxdy
when
D
is a general region in the
xy
plane the choice of order of integration can depend on a number
of factors. Since it’s always best to keep the integration as simple as possible, it’s always best to
consider which order is preferable.
Example 1:
in the case of
I
=
D
f
(
x, y
)
dxdy
where
D
is shown to the right, then fixing
x
and
integrating first with respect to
y
makes good
sense because then
D
=
(
x, y
) :
φ
(
x
)
≤
y
≤
ψ
(
x
)
,
a
≤
x
≤
b
for suitable choices of
a, b
and functions
φ
(
x
)
,
ψ
(
x
)
so that
D
f
(
x, y
)
dxdy
=
b
a
ψ
(
x
)
φ
(
x
)
f
(
x, y
)
dy
dx .
x
y
But if we had chosen to fix
x
, then the inte
gral with respect to
y
sometimes splits into two
parts as shown in red. This would make inte
gration with respect to
y
more complicated!
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Example 2:
similarly in the case of
I
=
D
f
(
x, y
)
dxdy
where
D
is shown to the right, then fixing
y
and
integrating first with respect to
x
makes good
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 Spring '07
 TextbookAnswers
 Factors, Integrals, dy dx, red line

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