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# 15-2 - Math 200 Spring 2010 Handout#18 Section 15-2 Double...

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Math 200, Spring 2010 Handout #18 Section 15-2 Double Integrals Over More General Regions Definition of Double Integral of f Over General Region D Suppose D is a bounded region enclosed in rectangular region R . If ( ) ( ) ( ) ( ) , if , , 0 if , f x y x y D f x y x y D = ɶ , then ( ) ( ) , , D R f x y dA f x y dA = ∫∫ ∫∫ ɶ We say that f is integrable over D if the integral of f ɶ over R exists. The value of the integral does not depend on the choice of R because f ɶ is zero outside of D. Approximating Double Integrals of f ( x , y ) over a glyph1197onrectangular Domain D by Riemann Sums. Choose a rectangle R containing D and subdivide R into Mglyph817 subrectangles i j R of size A x y Δ = Δ Δ . Choose a sample point i j P in each i j R . Since ( ) 0 i j f P = unless i j P lies in D , the Riemann sum reduces to ( ) ( ) ( ) , , 1 1 1 1 , , glyph817 M glyph817 M i j i j i j i j D glyph817 M x y x y S f P f P f x y dA = = = = = Δ Δ = Δ Δ ∑∑ ∑∑ ∫∫ ɶ 1. Let D be the shaded region in the figure at the right and let ( ) , f x y x y = . Approximate ( ) , D f x y dA ∫∫ by computing 4,4 S for ( ) , R f x y dA ∫∫ ɶ using the regular partition [ ] [ ] 0,2 0,2 R = × and the upper right-hand corners of the squares as sample points.

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15-2 - Math 200 Spring 2010 Handout#18 Section 15-2 Double...

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