sn12_1 - = = + = = = = ( , , ) ( , , ) where ( , , ) ( cos...

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Math 2433 – Notes week 12 (16.7-16.9) Popper012 1. What day is this? A. Monday B. Tuesday C. Wednesday D. Thursday 2. Which of the following will represent the volume of the solid bounded above by the plane z = y + 4, below by the xy -plane, and on the sides by the circular cylinder x 2 + y 2 = 16. a) ( ) 2 4 0 0 sin 4 r rdrd π θ + ∫ ∫ b) ( ) 2 4 0 0 sin r rdrd ∫ ∫ c) ( ) 4 0 0 sin 4 r rdrd + ∫ ∫ d) ( ) 2 4 0 0 sin 4 r drd + ∫ ∫
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16.7 Triple Integrals Integration over an arbitrary solid: Properties: 1. Linear 2. Order 3. Additivity Applications: 1. “ Volume” of “hypersolid” = ( , , ) S f x y z dxdydz ∫∫∫ 2. Volume of S = S dxdydz ∫∫∫ Reduction to a repeated integral 1. Type I: ( ) ( ) ( ) ( ) 1 2 1 2 , , a x b x y x x y z x y φ ψ ≤ ≤ ≤ ≤ ≤ ≤ 2. Type II: ( ) ( ) ( ) ( ) 1 2 1 2 , , c y d y x y x y z x y ≤ ≤ ≤ ≤ ≤ ≤ 3. Type III: 4. …… Examples: 1. Evaluate: 1 1 0 1 0 4 xy x x zdzdydx + - ∫ ∫ ∫
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Popper012 3. Evaluate the repeated integral: a) b) c) d) e)
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16.8 Cylindrical Coordinates 2 2 2 ( , , ) ( , , ) where cos sin tan x y z r z x r r x y y y r x z z z z θ
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Unformatted text preview: = = + = = = = ( , , ) ( , , ) where ( , , ) ( cos , sin , ) f x y z dxdydz F r z rdrd dz F r z f r r z = = Examples: Popper012 4. Give the integral using cylindrical coordinates to find: Where T : 0 < x < 1, 0 < y < , 0 < z < 2. a) ( ) 1 2 2 0 0 0 sin r rdzdrd b) ( ) /2 1 2 2 0 0 0 sin r rdzdrd c) ( ) 2 1 2 2 0 0 0 sin r rdzdrd d) ( ) 1 2 2 0 0 0 sin r dzdrd 16.9 Spherical Coordinates ( , , ) ( , , ) x y z where Triple Integrals Use when there is spherical symmetry Examples: Popper012 5. Find the spherical coordinates of the point with cylindrical coordinates : a) b) c) d) e) 6. Evaluate using spherical coordinates. Where T : 0 < x < 2, 0 < y < , < z < a) b) c) d) e)...
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This note was uploaded on 07/20/2010 for the course MATH 18427 taught by Professor Etgen during the Fall '10 term at University of Houston.

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sn12_1 - = = + = = = = ( , , ) ( , , ) where ( , , ) ( cos...

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