eget let fHisF Ipa

sass L pChetnik

FEDour unit ball B lip 9,0 off 1

Improper Integral

Lfo ftp.etcxihztdxdydzexists

where 135 1194,0 Oepee

Legio fix dxdy.dz

Legio I fo f fiftainoldpdoldohesiod do Siamoldd Kdp

gtfo 41T G E 41T

Hencewe saidthat f ftp.z.is integrableCuisense ofImproperIntegral

Fu gHy1

EYED

Lugo fff glyy dxdydz

BlBeJ elegio So ptsKbuilodpdfdo

legio 4Th log I to notexist

Hence wesaid that g is notintegrable

Question determine all p o suchthat

f pts is witgnable ni134123 ExD

Applicationofmaltiplenitefrats Thomas calculus 5156

In applications we often use the following

In Z din R is a region in115 with density 8

Firstmoment about y axis

My Sf x 84gsdA

First moment about X axis

Mx Sf y ou y DA IIMass M If any DA

Center of Maas centroid

cx55I Tf Th

In 3 date D solidregion in Raith axis.tt

Firstmomeuts

about YZ plane

Myz xoix.y.edu

about xz plane

Mxz you y dV

about xy planem ss.szax.us

dv

Mass fy axis DV

CenterofMass Centroid

cxiyizt CMF.MII.MY

I RrjmmRwMHxyMomentofinertiaabout the X axis I If504ydAaboutthe y axis Ig Sf Er DA

I

about ane

IL f rex yJocky dA

where ragy distance betweenCay and L

about the origin Io Sf y cragy dA

t.DNdrgmmkwmpxyzsmomeutsofiuert.ie

around x axis I µy F dV

aramdy axa Is do

around Z axis Iz city oxy DV

around line L IE rex y z5okay du

where ray D distance between Ky f and Li

egI consider D r Exty't E E R2with density ayy 8 Express It in

termof m mass of DEWICDr and R

r

sold Iz city's Idf

S S S photo Esinoldpdoldo

do singdol Hdp

21T Eg13552

9EUR73 r

Mass M G3 R o

Iz 2nFREIJI true for F t

There are two utterestaig special cases

d r o ie the whole solid ball

fIz2 R

di r R ie a ollow sphere made ofinfinitesimally thou sheet

Iz Lyi2 EEr

3 55 2h31

Moment of inertiathe solid ball assuming the same mass