Lectures

/Area⑨=ffpIrridA/surface area ( 16.6 ) -

surface integrals C 16.7 ) f-defined by Riemann

fffsfcxiyiztds.ee#ms

→ =ffDfCr7uoDlFuxFoldA•

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Interpretation :

ffgfcxlyizlds = Area (S ) .

. fAverage value ofFlair ) = sxcuio ) , your ) , zcuiv ) > f on S )

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ZooM_ : Example : 5- sphere of radius a

AA /AS Plato ) = Casino cosa ,

a Sino Sino,

a cos of )¥. ¥

.÷÷÷÷¥÷÷÷.

I

a→= Fcuotsu , ro ) - Flue , oo ) =/ a olcoso

acotsosino-

ashing/I LIU Luo , Vo ) DU - asinosinf a Sino co so 0

I = Fcuo, rotsv ) - Fluoro ) = of

sinffsinodosoi-sino-jst~ZILuoi.ro

) so cos 0kt )

f-area parallelogram try xPoI=a2 sin of ( Offs 't )

AS = lo-ixbl-lruxrolsus.ir

Area -

- ffpiroxrolda Example: S flat , containedin Xy - plane :

=L,

fo ! adsinododo F Cuco ) = Glued , yawl , 03.

=2ta2fjsino/d¢=4ta2¢ FuxrI=/£LIEto /-

IT If 0

Example : ffsxads = =¥afIo- FILE )Ix

'

ds- - -

=fofo!! Casino cos atYasin do 2Cn*Huo )

=a4fo⇐w5odAfjsin3o/do/ "Jacobian determinant

"

= . . . = 4zIa4¢-

Example : 5- graph of fcxiy )

over D

Flay ) = Cay , fcxiy ) )

* xrui.fiIf0 I fy

=itfx ) + Iffy ) the . I

-IF'xxr→yl=VCfxPtCfyPtl

Area Cst =ffpV¢xT-CfyT dxdy-

ds