Lecture19_Triple_integrals_and_3D_Coordinates_9-15

Lecture19_Triple_integrals_and_3D_Coordinates_9-15 -...

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11 Spherical Coordinates Naming convention : a point P(x,y,z) ) , , ( θ φ ρ ) / ( tan ), / ( , cos sin sin cos sin 2 2 1 2 2 2 z y x x y z y x x + = = + + = = = = tan z , y , 1 - Relation to Cartesian coordinates (switching) : ρ varies from 0 to ; φ varies from 0 to π θ varies from 0 to 2 π , Spherical coordinates are good for describing solids that are symmetric around the point.
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12 Spherical Coordinates sin , θ φ ρ d d d dV volume al differenti 2 = d sin d sin d a d d S t cons a d d S t cons a d d S t cons n dS S d surface al Differenti ˆ , tan ˆ , tan ˆ , tan ) ˆ ( 2 = = = = = = = r r r r
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13 Triple Integrals in Spherical Coordinates φ ρ θ d d d u I z y x Substitue d d d u dxdydz z y x f DD sin ) , , ( cos , sin sin , cos sin : sin ) , , ( ) , , ( 2 2 ∫∫ ∫∫∫ ∫∫∫ = = = = = - f i r st Æ l a s t express dV as
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14 Example 3 (P9-15.82) Find the moment of inertia about the z axis of the solid: Example 1 (P9-15-69): Convert the following equation to spherical coordinates: 2 2 2 3 3 y x z + = 2 2 2 2 a z y x = + + The density ρ v = k ρ . s Coordinate Spherical + = ∫∫∫ ; ) , , ( ) ( 2 2 V v z dxdydz z y x y x I ρ Example 2 (P9-15.76): Find the volume bounded by: Octant First , 0 , 3 , , 4 2 2 2 = = = = + + z x y x y z y x Give details of solutions
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15 [ ] [] θ φ ρ + + = = E E E E v sin 1 sin sin 1 1
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This note was uploaded on 05/09/2010 for the course ENGR 233 taught by Professor S.samuelli during the Winter '10 term at Concordia Canada.

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Lecture19_Triple_integrals_and_3D_Coordinates_9-15 -...

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