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# 15-3 - Math 200 Spring 2010 Handout#19 Section 15-3 Triple...

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Math 200, Spring 2010 Handout #19 Section 15-3 Triple Integrals Definition: Triple Integral Over a Rectangular Box Consider a function f of three variables defined on a rectangular box: [ ] [ ] [ ] ( ) { } 3 , , , , , | , , B a b c d p q x y z a x b c y d p z q = × × = . Divide interval [ ] , a b into l subintervals [ ] 1 , i i x x of equal width ( ) / x b a l Δ = , interval [ ] , c d into m subintervals 1 , j j y y of equal width ( ) / y d c m Δ = , and interval [ ] , p q into n subintervals [ ] 1 , k k z z of equal width ( ) / z p q n Δ = . This partitions box B into lmn sub-boxes [ ] [ ] 1 1 1 , , , i jk i i j j k k B x x y y z z = × × , each with volume V x y z Δ = Δ Δ Δ . Choose a sample point ( ) * * * , , i jk ijk ijk x y z in each i jk B and form the triple Riemann sum ( ) * * * 1 1 1 , , l m n ijk ijk ijk i j k f x y z V = = = Δ ∑∑∑ . The triple integral of ( ) , , f x y z over the box B is defined as ( ) ( ) * * * , , 1 1 1 , , lim , , l m n ijk ijk ijk B l m n i j k f x y z dV f x y z V →∞ = = = = Δ ∑∑∑ ∫∫∫ , if this limit exists Fubini’s Theorem for Triple Integrals If f is continuous on [ ] [ ] [ ] , , , B a b c d p q = × × , then the triple integral exists and is equal to the iterated integral: ( ) ( ) , , , , q d b p c a B f x y z dV f x y z dxdydz = ∫ ∫ ∫ ∫∫∫ To evaluate this iterated integral, we integrate first with respect to x (keeping y and z fixed), then we integrate with respect to y (keeping z

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