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