2006 Fall MATH 100 Lecture 81 MATH 100 Lecture 18 Triple Integral, II; centroid Class Triple...

2006 Fall MATH 100 Lecture 8 1

MATH 100 Lecture 18 Triple Integral, II; centroid

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and

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are orders Then the

),(),,(),(),,(

or

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Class Triple integral, centroidsIntegration in other orders



V

M

Center of gravity of a solid:Solid G homogeneous (composition & structure uniform)

Solid G inhomogeneous

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1



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region theof centroid have we),const a(),,( s,homogeneouFor

Explanation of center of gravity:

Consider a point-mass in located at x, then the tendency for the mass to produce a rotation about a point a on the axis is measured by the following quantity: moment of m



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assuming

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22

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dV(x, y, z)zdV(x, y, z)z

z

akhdxxakh

dxdyhk

dxdydzkz

dV(x, y, z)M

kz(x, y, z)

ah

const(x, y)

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a

a

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a

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a

a

a

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xa

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a

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h

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centroid by the travelleddistanceof area Volume

is about revolvingby format solid of volumethe

thenintersect,not line, a region, plane a :Pappus) of (Theorem :Thm

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LR



R) of (area 22

222

22

:Proof

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xdA

xdAdAΔAxV

ΔAxΔyΔxxΔV

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R

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n



222 22

revolvingby generated -torus- region,-circular - :Ex

brrbΔV

RVR

n

2006 Fall MATH 100 Lecture 81 MATH 100 Lecture 18 Triple Integral, II; centroid Class Triple...

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