15-1 - Math 200 Spring 2010 Handout#17 Section 15-1...

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Math 200, Spring 2010 Handout #17 Section 15-1 Integration in Several Variables Definition: Double Integral Over a Closed Rectangle Consider a function f of two variables defined on a closed rectangle [ ] [ ] ( ) { } 2 , , , | , R a b c d x y a x b c x d = × = ≤ ≤ ≤ ≤ . Divide interval [ ] , a b into m subintervals [ ] 1 , i i x x of equal width ( ) / x b a m Δ = and interval [ ] , c d into n subintervals [ ] 1 , i i y y of equal width ( ) / y d c n Δ = . This partitions rectangle R into nm subrectangles [ ] [ ] 1 1 , , i j i i i i R x x y y = × , each with area A x y Δ = Δ Δ . Choose a sample point ( ) * * , i j ij x y in each i j R . If ( ) , 0 f x y , then the volume of the rectangular “column” with base i j R and height ( ) * * , i j i j f x y is ( ) * * , i j i j i j V f x y A = Δ . The volume of the solid that lies under the graph of f and above rectangle R is approximated by a double Riemann sum ( ) * * 1 1 , m n ij ij i i V f x y A = = Δ ∑∑ . In the limit as , m n →∞ , this volume is defined as ( ) , * * 1 1 lim , m n m n ij ij i i V f x y A →∞ = = = Δ ∑∑ The double integral of ( ) , f x y over a rectangle R is defined as the limit ( ) ( ) , * * 1 1 lim , , R m n m n ij ij i i f x y dA f x y A →∞ = = = Δ ∫∫ ∑∑ If this limit exists, we say that ( ) , f x y is integrable over R .
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