2006 Fall MATH 100 Lecture 81 MATH 100 Lecture 19 Triple Integral in cylindrical & spherical...

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2006 Fall MATH 100 Lecture 8 1 MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate height wedge the of thickness angle central ) ( , ) ( , ) ( , 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 1 2 1 z z r r z z z z z z r r r r r r Class 19 Triple Integral in cylindrical & spherical coordinate Define cylindrical wedge or cylindrical elements of volume
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Transcript of 2006 Fall MATH 100 Lecture 81 MATH 100 Lecture 19 Triple Integral in cylindrical & spherical...

Page 1: 2006 Fall MATH 100 Lecture 81 MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate Class 19 Triple Integral in cylindrical & spherical.

2006 Fall MATH 100 Lecture 8 1

MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate

height

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angle central

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Class 19 Triple Integral in cylindrical & spherical coordinate

Define cylindrical wedge or cylindrical elements of volume

Page 2: 2006 Fall MATH 100 Lecture 81 MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate Class 19 Triple Integral in cylindrical & spherical.

2006 Fall MATH 100 Lecture 8 2

MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate

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Page 3: 2006 Fall MATH 100 Lecture 81 MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate Class 19 Triple Integral in cylindrical & spherical.

2006 Fall MATH 100 Lecture 8 3

MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate

solid simplefor Evaluation

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),(),,(),(),,( 21

R

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Page 4: 2006 Fall MATH 100 Lecture 81 MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate Class 19 Triple Integral in cylindrical & spherical.

2006 Fall MATH 100 Lecture 8 4

MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate

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Page 5: 2006 Fall MATH 100 Lecture 81 MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate Class 19 Triple Integral in cylindrical & spherical.

2006 Fall MATH 100 Lecture 8 5

MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate

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Page 6: 2006 Fall MATH 100 Lecture 81 MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate Class 19 Triple Integral in cylindrical & spherical.

2006 Fall MATH 100 Lecture 8 6

MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate

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Page 7: 2006 Fall MATH 100 Lecture 81 MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate Class 19 Triple Integral in cylindrical & spherical.

2006 Fall MATH 100 Lecture 8 7

MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate

4

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x

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Page 8: 2006 Fall MATH 100 Lecture 81 MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate Class 19 Triple Integral in cylindrical & spherical.

2006 Fall MATH 100 Lecture 8 8

MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate

)(,

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2121

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Page 9: 2006 Fall MATH 100 Lecture 81 MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate Class 19 Triple Integral in cylindrical & spherical.

2006 Fall MATH 100 Lecture 8 9

MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate

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sin

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Page 10: 2006 Fall MATH 100 Lecture 81 MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate Class 19 Triple Integral in cylindrical & spherical.

2006 Fall MATH 100 Lecture 8 10

MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate

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Page 11: 2006 Fall MATH 100 Lecture 81 MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate Class 19 Triple Integral in cylindrical & spherical.

2006 Fall MATH 100 Lecture 8 11

MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate

)22(2

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Page 12: 2006 Fall MATH 100 Lecture 81 MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate Class 19 Triple Integral in cylindrical & spherical.

2006 Fall MATH 100 Lecture 8 12

MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate

du

dxuxf

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Page 13: 2006 Fall MATH 100 Lecture 81 MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate Class 19 Triple Integral in cylindrical & spherical.

2006 Fall MATH 100 Lecture 8 13

MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate

ation transform theofJacobian theis

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vuyyvuxx

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xyvuyyvuxx

uv

Page 14: 2006 Fall MATH 100 Lecture 81 MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate Class 19 Triple Integral in cylindrical & spherical.

2006 Fall MATH 100 Lecture 8 14

MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate

),(

),(lim

VU),(),,(lim),(),,(

limit The

VU),(),,( sum partial From

VUkA

then

,VU area of plane-uvin rectanularth -i ingcorrespond the, A area with A as

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uv

Page 15: 2006 Fall MATH 100 Lecture 81 MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate Class 19 Triple Integral in cylindrical & spherical.

2006 Fall MATH 100 Lecture 8 15

MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate

brarrrR

rrrrR

),()(, :II Type

),()(, :I Type

21

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rJ

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sin,cos

:Ex

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rdrdrrfdxdyyxfRRxy

Page 16: 2006 Fall MATH 100 Lecture 81 MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate Class 19 Triple Integral in cylindrical & spherical.

2006 Fall MATH 100 Lecture 8 16

MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate

drfdrdrdrf

drdrfdrdrf

b

aR

R

or

Then

xyuv

xyxy

R

RR

xxu

CxCx

RRdxdyyxxy

one toone 1,24,1then

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equations having

linesstraight are of sides The .over Evaluate :Ex

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3

Page 17: 2006 Fall MATH 100 Lecture 81 MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate Class 19 Triple Integral in cylindrical & spherical.

2006 Fall MATH 100 Lecture 8 17

MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate

3

1

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1

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333

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Page 18: 2006 Fall MATH 100 Lecture 81 MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate Class 19 Triple Integral in cylindrical & spherical.

2006 Fall MATH 100 Lecture 8 18

MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate

wvuzwvuywvuxfwvuf

dudvdwwvu

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uvwxyz RR

,,,,,,,,,, where

,,

,,,,,,

tionGeneraliza

Page 19: 2006 Fall MATH 100 Lecture 81 MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate Class 19 Triple Integral in cylindrical & spherical.

2006 Fall MATH 100 Lecture 8 19

MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate

sin)sin(cossin

cossin

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cos

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nsApplicatio

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xxx

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z

y

x

Page 20: 2006 Fall MATH 100 Lecture 81 MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate Class 19 Triple Integral in cylindrical & spherical.

2006 Fall MATH 100 Lecture 8 20

MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate

dzrdrdfdxdyxzf

rr

r

J

zz

ry

rx

zxyz RR

100

0cossin

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sin

cos

)2(