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lecture notes 16.6-16.9 (1)

# lecture notes 16.6-16.9 (1) - 16.6(15.6 Surface areas A...

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1 16.6 (15.6) Surface areas A surface S can be described by parametric equations: ( ) ⟨ ( ) ( ) ( )⟩ ( ) (S is called a parametric surface.) Examples (1) The graph of ( ) : ( ) ⟨ ( )⟩ In cylindrical coordinates, the graph of ( ) : ( ) ⟨ ( )⟩ In spherical coordinates the graph of ( ) : ( ) ⟨ ( ) ( ) ( )

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2 (2) ( ) ⟨ , or Is the cylinder: (3) ( ) , represents: (4) The sphere can be parameterized as , .
3 Consider the surface S: ( ) ( ) . Formula: Area of S = | | Remark: In particular, if S the graph of ( ) ( ) , then the area of S is dA

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4 Examples (1) Find the area of the ellipse cut from the plane By the cylinder . (2) Find the area of the surface that lies under the plane
5 (3) Find the area of a sphere of radius .

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6 16.7 Surface integral Suppose S: ( ) ( ) . Then the surface integral of a function over S is ( ) ( ( )) | | Remark: (1) ( ) . (2) Suppose S is the graph of ( ) . Then ( ( ))√
7 Examples (1) Suppose S is the cylinder . Evaluate (2) Let S: . Evaluate .

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8 Suppose a surface S: ( ) ( ) , is oriented (i.e., for

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lecture notes 16.6-16.9 (1) - 16.6(15.6 Surface areas A...

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