13.2 DoubleIntegrals

13.2 DoubleIntegrals - 1 3 .2 
D o u b l e 
I n t e g r...

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Unformatted text preview: 1 3 .2 
D o u b l e 
I n t e g r a l s 
o v e r 
G e n e r a l 
R e g io n s 
 Suppose
the
region
is
not
rectangular.

Integrate
 π 2 cosθ 0 0 ∫∫ e sin θ drdθ 
 
 
 
 
 
 
 
 
 
 
 
 Def:

An
xy
region
(or
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
region
can
be
described
by
a
set
 of
inequalities

a
≤
x
≤
b
and
h(x)
≤
y
≤
g(x).
 
 
 gHxL hx a 
 b 
 Def:

An
xy
region
(or
xy
plane)
is
called
a
type
II
region
if
any
 horizontal
strip
(in
the
x
direction)
always
has
the
same
left
 and
right
boundaries
and
the
region
can
be
described
by
a
set
 of
inequalities

c
≤
y
≤
d
and
h(y)
≤
x
≤
g(y).
 
 c hHyL gHyL d 
 
 
 b g(x ) ∫∫ If
type
II
use

 ∫ ∫ If
type
I

use
 f ( x, y ) dydx 
 a h(x ) d g( y ) c h(y ) f ( x, y ) dxdy 
 
 
 Ex.

Suppose
R
is
bounded
by
y
=
x2
and
x
+
y
=
6.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Suppose
R
is
the
triangular
region
bounded
by
(1,
3),
(2,
1)

 and
(4,
4).
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Sketch
the
regions
of
integration
and
reverse
the
order
of
 integration.
 
 63 Ex.
 
 
 
 
 
 
 
 ∫ ∫ f ( x, y ) 01 4 Ex.
 
 
 
 
 
 
 
 
 0 0 e Ex.
 
 
 
 
 
 
 
 y ln x ∫∫ ∫∫ 1 0 dydx 
 f ( x, y ) dxdy 
 e y dydx 
 1 Ex.
 
 
 
 
 
 
 
 
 
 ∫∫ y +1 −1 − y +1 f ( x, y ) dxdy 
 π Ex.

Integrate
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 π 0 x ∫∫ sin y dydx 
 y Do:
Sketch
the
region
and
reverse
the
order
of
integration
for
 1 2.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ∫∫ 2 1.

 
 y y2 0 y2 ∫∫ 1 y f ( x, y ) dxdy 
 f ( x, y ) dxdy 
 Ex.

Find
the
volume
of
the
solid

whose
base
is
the
region
in
 the
xy‐plane
that
is
bounded
by
the
parabola
y
=
4
–
x2
and
the
 line
y
=
3x,
while
the
top
of
the
solid
is
bounded
by
the
plane

 z
=
x
+
4.

 ...
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