14.6 Notes - Calculus III Section 14.6 Surface Integrals z...

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Calculus III – Section 14.6 – Surface Integrals Let S be a surface with equation ) , ( y x g z = and let R be its projection on the xy-plane. If x g g , and y g are continuous on R and f is continuous on S, then the surface integral of f over S is ( 29 ∫ ∫ ∫ ∫ + + = R y x S dA y x g y x g y x g y x f dS z y x f 2 2 )] , ( [ )] , ( [ 1 ) , ( , , ) , , ( . Example: Evaluate ∫ ∫ + - S dS z y x ) 2 ( where 4 0 , 2 0 , 10 2 2 : = + - y x z y x S For a surface S given by the vector-valued function k v u z j v u y i v u x v u r ) , ( ) , ( ) , ( ) , ( + + = defined over a region D in the uv-plane, the surface integral of ) , , ( z y x f over S is given by ( 29 ∫ ∫ ∫ ∫ × = D v u S dA v u r v u r v u z v u y v u x f dS z y x f ) , ( ) , ( ) , ( ), , ( ), , ( ) , , ( Example: Evaluate ∫ ∫ S dS y x f ) , ( where π = + = v u u v u v u v u r S y x y x f 0 , 4 0 ; , sin 2 , cos 2 ) , ( : ; ) , ( .
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A surface is called orientable if a unit normal vector
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14.6 Notes - Calculus III Section 14.6 Surface Integrals z...

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