ch 8- multiple integrals
TRANSCRIPT
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CH 8-Multiple Integrals
Recall (definite integral)
=
n
kk
n
xxf
1
)(lim
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1. Double
Integrals
iii Ayx ),(
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n = 16n = 64
n = 256
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Properties of Double Integrals
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2.Evaluation of * *
(1) Rectangular regions
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The above result wasproved by Italianmathematician Fubini
(1907) under the
condition thatf(x,y) is
continuous throughout
the regionR.
Guido Fubini
(1879-1943)
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Remark. If f(x,y) =g(x)h(y), then
(*)
(*)
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How to evaluate
( , )
efficiently ?
R f x y dA
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General regions Type A
The regionR :
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General regions Type B
The regionR :
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()
Find xy
dA
,
(Type A) 1. Sketch
2.y-limits
3.x-limits
()
=
=
= = 1
24.
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()
(Type B) 1. Sketch
2.x-limits
3.y-limits
()
=
=
=
(y
y)dy
= 1
24
Find xy
dA
,
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()
Evaluate whereR :
(Type B)
() = ???
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()
Evaluate whereR :
(Type A)
()
=
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()
Evaluate
(1) Type A orB ? (B)
(2) IdentifyR
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()
=
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3.Double Integral
inPolar Coordinates
CircleR :
Sector of a circleR :60o
1
R
y
x
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Ring
R :
Polar rectangle
R :
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Change ofvariables
(x,y) (r, ),
?
(x,y)
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IfR :
then
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Evaluate
R :
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Applications
VolumeSuppose
D is a
solid
under the surfacef(x,y)over a plane regionR.Then the volume ofD
is given by
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Elliptic paraboloid
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The
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Surface area
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2 2
2
4
4
x
x
+
1sin6 2
=
2 2
2
4
4
x x
x
+
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Mass & center of gravity
If a lamina with a continuous density fn (x,y)
occupies a regionR in thexy-plane, its total
mass M is given by& its center of gravity is
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Find the center ofgravity of the triangularlamina with vertices (0,0),(0,1) & (1,0), & density
function (x,y) =xy.
(0,1)
(1,0)
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For the center of gravity,
Thus
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5. Triple Integral
Recall :
=
n
kk
nxxf
1)(lim
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iii Ayx ),(
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f(x,y,z) is defined onD
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Physical Meaning of
--------------
No direct geometrical meaning for
If
Iffrepresents certain physical quantity, thenmay have some physical meaning.
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Now, suppose thedensity is a fn (x,y,z) defined
onD.
& so
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Evaluation of triple integral
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() Evaluate
() =
=
=
= = 1