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Multivariable Ch 12 Review

# Multivariable Ch 12 Review - Mayur Pandya Multivariable...

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Mayur Pandya 4/23/2006 Multivariable Calculus Chapter 12 Review 12.1: Double Integrals over Rectangles 1) Single Integration: = n i i n x x f 1 * ) ( lim = b a dx x f ) ( 2) Definition of a double integral: A y x f ij m i n j ij n m ∑∑ = = ) , ( lim * 1 1 * , = ∫ ∫ R dA y x f ) , ( a. Domain is rectangular 3) Double Riemann sum: A y x f ij m i n j ij ∑∑ = = ) , ( * 1 1 * a. Upper left corner b. Upper right corner c. Lower left corner d. Lower right corner 4) Average Value: Single variable function - b a dx x f a b ) ( 1 5) Average Value: Two variable function ∫ ∫ R dA y x f R A ) , ( ) ( 1 12.2: Iterated Integrals 1) dx dy y x f dydx y x f b a d c b a d c ∫ ∫ ∫ ∫ = ) , ( ) , ( dy dx y x f dxdy y x f d c b a d c b a ∫ ∫ = ) , ( ) , ( 2) Fubini’s Theorem = ∫ ∫ R dA y x f ) , ( ∫ ∫ = b a d c dydx y x f ) , ( ∫ ∫ d c b a dxdy y x f ) , ( IF f is bounded on R, f is discontinuous only on a finite number of smooth curves, and its integrated integrals exist. 12.3: Double Integrals over General Regions 1) Type I Regions ∫ ∫ ∫ ∫ = D b a x g x g dydx y x f dA y x f ) ( ) ( 2 1 ) , ( ) , (

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Mayur Pandya 4/23/2006 Multivariable Calculus Chapter 12 Review 2) Type II Regions ∫ ∫ ∫ ∫
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Multivariable Ch 12 Review - Mayur Pandya Multivariable...

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