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UNIT 4
MULTIPLE
INTEGRALS
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Double integral
Given: ( )y , x f z=
( )
∫∫ Rd y , x f is evaluated
over
a region on the xy -plane.
R
here dA is either dydx ordxdy .
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Double integral as area
!onsider a region on the xy - plane.
R
∫∫ R
d gives the area o"the region.
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Example. Set-up the iterated that ill
give the area o" the region #ounded#$ and .
Solution:
% &
%
&
'
( y &
&
x y =
2 x y =
x y 2
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Using as order o" integrationdydx
Solution
) verti*al strips
dx dy
∫ ∫
0
2
2 x
x 2
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Using as order o" integrationdx dy
Solution
) hori+ontal strips
dydx
∫ ∫
0
4
2
y
y
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Double integral as volume
Let #e a height o" are*tangular #o, o" a solid over a
region .R
gives the volue
o" the solid.
( )y , x f z=
( )
∫∫ Rd y , x f
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Triple integrals over solids
Let #e de"ined and*ontinuous/ over a solid .
z ,y , x f w=
∫∫∫
S
d z ,y , x f
here has si, possi#le orders
o" integration.
dV
The triple integral o" over a solid
in is given #$
f ' R
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Triple integrals as volume
∫ ∫ ∫
S
d gives the volueo" the solid .
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Example. Set-up the iteratedintegral that solves "or the
volue
o" the tetrahedron #ounded #$ the
plane and the
*oordinate planes.
0 1 & '
= zy x
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% &
%
&
'
R
∫ ∫
R
d y x & ' 1
∫ ∫
R
d y , x hV
0 1 & ' =y x
d dy y x ∫ ∫
& ' 1
0
&
0
& ' 1
z 0
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Example. Set-up the iteratedintegral that solves "or the
volue
o" the solid in the "irst o*tant
#ounded #$ the para#oloid
and the *$linder.
& & y x z
( & & =
y x
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(
&
&
& & y x z
& &
y x y , x h
(
& & =
y x
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% &
%
&
R
∫ ∫
R
d y x & &
∫ ∫
R
d y , x hV
(& & =y x
d dx y x ∫ ∫
& &
0
&
0
& ( y
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Example. Set-up the iterated tripleintegral that solves "or the
volue
o" the solid #ounded #$ the
*$linder 2 the plane
and the plane.
&3 & & =
y x
4 z x xy
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∫ ∫ ∫
S
d V
∫ ∫ ∫
d dy dz
0
4
& &3 x
& &3 x
3
3 &3
& &
=
y x
4 z x
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Example. Set-up the iterated triple
integral that solves "or the volue
o" the solid #ounded #$ the
para#oloids and .
& &
y x z
& & 1 y x z
These para#oloids interse*t at .' z
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∫ ∫ ∫
S
d V
∫ ∫ ∫
dxd dz'
& &
y x z
& & 1 y x z
& &
y x
& & 1
y x
' & & =y x
& ' y
& ' y
'
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