13.5 Notes - x y x y x R tan , 4 : , : . The top 2 graphs...

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Calculus III Section 13.5 – Surface Area When finding surface area of a solid we will be extending the formulas that we used back in Chapter 9 to functions of more than one variable just as we have done for other applications. Definition of Surface Area If f and its first partial derivatives are continuous on the closed region R in the xy -plane, then the area of the surface S given by ) , ( y x f z = over R is given by Surface area [ ] [ ] ∫ ∫ ∫ ∫ + + = = R y x R dA y x f y x f dS 2 2 , ( ) , ( 1 Example: (#6) Find the area of the surface given by 2 ) , ( y y x f = over the region in the plane bounded by R: square with vertices ( 29 ( 29 ( 29 ( 29 3 , 3 , 3 , 0 , 0 , 3 , 0 , 0 Graphs of 2 ) , ( y y x f = over a general region and the region indicated in the problem are given below.
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Example: (#9) Find the area of the surface given by x y x f sec ln ) , ( = over the region in the plane bounded by ( 29
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Unformatted text preview: x y x y x R tan , 4 : , : . The top 2 graphs below are of over a general region and the bottom graph is with our restriction in the plane. The red graph is of x y x f sec ln ) , ( = and the purple is x y tan = . Example: Find the area of the surface that is the portion of the paraboloid 2 2 16 y x z--= that is in the first octant. (#16) Example: Find the surface area of the portion of the cone 2 2 2 y x z + = (blue) that is inside the cylinder 4 2 2 = + y x (pink). (#18) Example: Set up the iterated integral to find the area of the surface that is the portion of the paraboloid 2 2 3 y xy x z--= over the region ( 29 { } x y x y x R , 4 : , : . (#30) The surface shown in red is 2 2 3 y xy x z--= ....
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This note was uploaded on 01/13/2012 for the course MATH 2574 taught by Professor Pamelasatterfield during the Spring '12 term at NorthWest Arkansas Community College.

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13.5 Notes - x y x y x R tan , 4 : , : . The top 2 graphs...

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