3213 Discussion Weeks

Volume of 3D regionProjection on my plane D

heightfunction EICX.gs EZE Zayz 224y

hexy i ZIYI Zi Xin

f Dp hex Nd'tdouble integral

yi Zuxyl

Compare with surface are

Sp hdr fab hirta at'andt

F 3D boundary carvescalar line integral

Changeof variables

Effigy3 Jacobian uh Ezn Ian.ie same

Changeof variables formulaG Do D fix y on D

p fixg dxjdf ffpofcxcuis.yeu.nl IYy dudV

dxdynl3Y'u1dudV areaofsmallpieces

Example polar coordinate

fit dodyerdrdd

r x

Nervally mo

ios rate

sino ruse f ri dxdynrdrdo

Sometimes weget µ u Fray insteadofnay _CHUN then

J.cat EiiTyl fYi.Yg It JcF5Similar for 3D Triple integral

WorkshutproblemsI

1

Eitiperez

i Dflx.pl xIyreYDo_Lino toerag

Heightfunction hcrf zr

p.itffphdAffp.hcr7rdrdp

changeof variablesdlt rdrdd

foudofoczr brdr

2n.f r3 fr831T

2 D LIExZyk4 lexyey xzayzoChange of variables Yn int n

Tf 11 21 45724 D g

dxdy Jdudv2 y7 Hgs Gcuv

i ffp Hy'dA ftp.xkyh zT dudv

tz1Do 2 3.3 I

3 Application to mechanics

qYSI rampdylidx 111111xuxgtzctx45fxdxy x

J.IE3 x0x7iExlodx

zHotIF Iz

4 Application to probabilityla p lxtfe 23 fodxffxpx.ycx.gldy

yaf e it dx

teat azeez q1 Itaz e At o

cb p qcz de acuate AZ

Eze at

TETHfothzpxeyczidzpnotherwaytudoit

fo Ee

itdz.isbydowbkhtegrd

fotozize

XZdtjodx.fi21T e Hdz

Zp