lecture19

# Lecture19 - MATH 100 Lecture 19 Triple Integral in cylindrical spherical coordinate Class 19 Triple Integral in cylindrical spherical coordinate

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2006 Fall MATH 100 Lecture 8 1 MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate height wedge the of thickness angle central ) ( , ) ( , ) ( , 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 1 2 1 < = = < = = < = = z z r r z z z z z z r r r r r r θθ Class 19 Triple Integral in cylindrical & spherical coordinate Define cylindrical wedge or cylindrical elements of volume

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2006 Fall MATH 100 Lecture 8 2 MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate () {} 12 *** 1 Definiton of triple integral Step 1: cut into cylindrical wedges, labelled by , ,..., , Step 2: pick points , , in Step 3: Form the sum , , Step 3 n kkk k n k k GV V V rz V fr z V θ = Δ ΔΔ Δ Δ n 1 : Take limit lim , , , , n k k G fr zd V θθ →+∞ = Δ= ∫∫∫
2006 Fall MATH 100 Lecture 8 3 MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate solid simple for Evaluation { } region polar simple ) , ( ), , ( ) , ( ) , , ( 2 1 = R R r r g z r g z r G θθ

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2006 Fall MATH 100 Lecture 8 4 MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate () 2 1 22 2 11 1 (,) *** Thm: simple solid, simple polar region, then ,, = , , Rough proof: choose , , gr GR rg r r kkk f r z dV f r z dz dA f r z rdzdrd rz θ θθ −− ⎡⎤ = ⎢⎥ ⎣⎦ ∫∫∫ ∫∫ ∫ ∫∫ [] [ ] () () k k * kk k k the center of V V area of base A z r z It follows that l im , , V z A k nn n G g R height r fr zd V fr z z zd A fr zr →∞ == Δ Δ= ⋅Δ = Δ Δ ⋅Δ = Δ Δ ∑∑ rr r dzdrd
2006 Fall MATH 100 Lecture 8 5 MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate () {} 22 2222 235 00 0 3 23 3 / 2 0 Ex:find the volume & centroid bounded by 5 9 0 Sol: ,, 0 5 , 0 3 , 0 2 12 52 ( 5 ) r G xyz xy z Gr z z r r V dV rdzdrd rr d r d r π θ θπ ++= += = =≤ == =⋅ = = ∫∫∫ ∫ ∫ ∫ ∫∫ 32 2 3 / 2 3 3 2 122 5( 53 ) 54 33 3 ππ ⎡⎤ −− = = ⎣⎦

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2006 Fall MATH 100 Lecture 8 6 MATH 100 Lecture 19 Triple Integral in cylindrical & spherical coordinate () 0 symmetry By 488 1107 4 3 2 9 25 122 3 4 1 2 1 25 2 244 3 5 122 3 2 1 122 3 122 3 1 4 3 0 4 3 0 2 2 0 3 0 2 2 2 0 3 0 5 0 2 2 0 3 0 5 0 2 2 2 2 = = =
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## This note was uploaded on 09/17/2011 for the course MATH 100 taught by Professor Qt during the Fall '09 term at HKUST.

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Lecture19 - MATH 100 Lecture 19 Triple Integral in cylindrical spherical coordinate Class 19 Triple Integral in cylindrical spherical coordinate

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