2433-Ch16-slides - Double Integrals f (x, y ) dxdy m n =...

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Double Integrals ±± Ω f ( x, y ) dxdy = lim m,n →∞ m ² i =1 n ² j =1 m ij Δ x i Δ y j = lim m,n →∞ m ² i =1 n ² j =1 M ij Δ x i Δ y j where m ij = min f ( x,y ) and M ij = max f ( x, y )o n R ij = lim m,n →∞ m ² i =1 n ² j =1 f ( x i ,y j x i Δ y j ( x 1 j ) an arbitrary point in R ij 1
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Properties I. Linearity ±± Ω ( αf + βg ) dxdy = α Ω f dxdy + β Ω g dxdy II. Additivity Ω f ( x, y ) dxdy = Ω 1 f ( x, y ) dxdy + Ω 2 f ( x, y ) dxdy 2
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III. Order If f ( x, y ) 0 on Ω, then ±± Ω f ( x, y ) dxdy 0 If f ( x, y ) g ( x, y ) on Ω, then Ω f ( x, y ) dxdy Ω g ( x, y ) dxdy 3
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Some Applications A. If f ( x, y ) 0 on Ω, then ±± Ω f ( x, y ) dxdy = volume of the solid which has Ω as its base, vertical sides, and the surface z = f ( x, y ) as its top. B. Ω 1 dxdy = areaΩ 4
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Double integrals/Repeated Inte- grals A. Ω = Type I region: ϕ 1 ( x ) y ϕ 2 ( x ) ,a x b ±± Ω f ( x, y ) dxdy = ± b a ² ± ϕ 2 ( x ) ϕ 1 ( x ) f ( x, y ) dy ³ dx = ± b a ± ϕ 2 ( x ) ϕ 1 ( x ) f ( x, y ) dy dx 5
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Example 1: f ( x, y )=3 x 2 +2 y on Ω: 0 x 2 ,x 2 y x . Calculate ±± Ω f ( x, y ) dxdy . Ω f ( x, y ) dxdy = ± 2 0 ± x +2 x 2 (3 x 2 y ) dy dx = 316 15 6
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B. Ω = Type II region: ψ 1 ( y ) x ψ 2 ( y ) ,c y d ±± Ω f ( x, y ) dxdy = ± d c ² ± ψ 2 ( y ) ψ 1 ( y ) f ( x, y ) dx ³ dy = ± d c ± ψ 2 ( y ) ψ 1 ( y ) f ( x, y ) dx dy 7
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Example 2: f ( x, y )= xy + 3 on Ω: y 2 x 2 y, 0 y 2 Calculate ±± Ω f ( x, y ) dxdy . Ω f ( x, y ) dxdy = ± 2 0 ± 2 y y 2 ( xy +3) dx dy = 20 3 8
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3. Interchange the order of integra- tion in Example 2. ±± Ω f ( x, y ) dxdy = ± 2 0 ± 2 y y 2 ( xy +3) dx dy = ± 4 0 ± x x/ 2 ( xy +3) dy dx = 20 3 9
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4. Interchange the order in Example 1. ± 2 0 ± x +2 x 2 (3 x 2 +2 y ) dy dx = ± 2 0 ± y 0 (3 x 2 y ) dx dy + ± 4 2 ± y y - 2 (3 x 2 y ) dx dy 10
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5. f ( x, y )= e - y 2 on Ω the region bounded by the y -axis, the line y = 1, and the line y = 1 2 x . Calculate ±± Ω f ( x, y ) dxdy . Ω f ( x, y ) dxdy = ± 2 0 ± 1 x/ 2 e - y 2 dy dx = ± 1 0 ± 2 y 0 e - y 2 dx dy =1 - 1 e 11
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6. Calculate ± 2 0 ± 4 x 2 2 x cos( y 2 ) dy dx by interchanging the order of inte- gration. ± 2 0 ± 4 x 2 2 x cos( y 2 ) dy dx = ± 4 0 ± y 0 2 x cos( y 2 ) dx dy = 1 2 sin(16) 12
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7. Find the volume of the solid bounded by the coordinate planes and the plane x 2 + y 3 + z 4 =1 V = ± 2 0 ± - 3 2 x +3 0 4 ² 1 - x 2 - y 3 ³ dy dx =4 13
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8. Find the volume of the solid bounded above by the paraboloid z = x 2 + y 2 , below by the x, y -plane, and on the sides by the cylinder x 2 + y 2 =4 . V = ± 2 - 2 ± 4 - x 2 - 4 - x 2 ( x 2 + y 2 ) dy dx ± 2 0 ± 4 - x 2 0 ( x 2 + y 2 ) dy dx = ··· 14
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B. Double integrals; Polar Co- ordinates Given F = F ( r, θ ) continuous on the polar rectangle Γ: a r b, α θ β.
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This note was uploaded on 05/19/2010 for the course MATH 18427 taught by Professor Etgen during the Spring '10 term at University of Houston.

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2433-Ch16-slides - Double Integrals f (x, y ) dxdy m n =...

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