MAT 1236
Calculus III
Section 15.3 Part II
Double Integrals Over
General Regions
http://myhome.spu.edu/lauw
HW&…
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WebAssign 15.3 II
(7 problems, 51 min.)
Quiz: 15.1-15.3
Preview
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Change the order of integration
Properties (relation to section 6.1)
Order of Integration
If D is both type I & type II, then

D
 b g2 ( x)
  f ( x, y )dydx
 a g1 ( x )
f ( x, y )dA   h ( y )
d 2

  f ( x, y )dxdy
 c h1 ( y )
Example 1
Evaluate
1 1
2
sin(
y
)dydx

0 x
Example 1
1 1
2
sin(
y
)dydx

0 x
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The inner integral is impossible/difficult
to evaluate
Switch the order of integration
Example 1
Sketch and state the region of integration
and change the order of integration
1 1
2
sin(
y
)dydx

0 x
Example 1
Sketch and state the region of integration
and change the order of integration
1 1
2
sin(
y
)dydx

0 x
Type I
Example 1
Sketch and state the region of integration
and change the order of integration
y 1
1 1
2
sin(
y
)dydx

0 x
yx
Example 1
Sketch and state the region of integration
and change the order of integration
x 1
1 1
2
sin(
y
)dydx

0 x
x0
1 1
Example 1
  sin( y
2
)dydx
0 x
1
(1  cos1)
2
Summary
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Notice that the inner integral is difficult to
evaluate
Sketch the region
Reverse the order of integration
Evaluate the iterated integral
Expectations
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You are expected to label your graph
Transitional integral
1 1
2
2
sin(
y
)
dydx

sin(
y
) dA


0 x
D
1 y
   sin( y 2 ) dxdy
0 0

You are expected to write down the two
set descriptions of the domain D.
Properties
 [ f ( x, y)  g ( x, y)] dA  f ( x, y) dA  g ( x, y) dA
D
D
 cf ( x, y) dA  c f ( x, y) dA
D
D
and others….
D
Property
1 dA 
area of D
D
z 1
A D
Consistency
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How do we know our calculus “system”
is consistent? (No conflicts, agree with
each other)
Consistency
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How do we know our calculus “system”
is consistent? (No conflicts, agree with
each other)
Volume formula “implies” area formula
via
1 dA  area of D
z 1
D
A D
Type I:
Region Bounded Above/Below
A( D )   1dA
y  g2 ( x)
D
b g2 ( x )


1dydx
a g1 ( x )
D
b
   y  y  g2( x ) dx
yg ( x)
1
a
a
b
b
   g 2 ( x)  g1 ( x)  dx
a
y  g1 ( x)
Type II:
Region Bounded Left/Right
x  h1 ( y)
A( D )   1dA
D
d
d h2 ( y )


1dxdy
c h1 ( y )
d
D
   x x  h ( y ) dy
x  h2 ( y )
1
c
d
   h2 ( x)  h1 ( x )  dy
c
x  h2 ( y)
c