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II. Integration – 1. Dobbelt integral 172 Hjørner (1,1),(2,1),(1,2),(2,2). y 2 1 1 2 1.15. Riemann sum til beregning ☞ [S] 12.1 Double integrals over rect. . . Eksempel 1 - fortsat Hjørner (1,1),(2,1),(1,2),(2,2) og ∆A = 1. Den dobbelte Riemann sum giver R = [0,2] × [0,2], f(x,y) = 16 − x 2 − 2y 2 ≥ 0 (16 − x R 2 − 2y 2 )dA ≈ f(1,1) + f(1,2) + f(2,1) + f(2,2) = 13 + 7 + 10 + 4 = 34 x
II. Integration – 1. Dobbelt integral 173 1.16. Test dobbelt integralet Test ☞ [S] 12.1 Double integrals over rectangles Lad f(x,y) = xy, R = [1,2] × [1,2]. Et skøn med en dobbelt Riemann sum giver for integralet V = xy dA R (a) V < 0. (b) V = 0. (c) V > 0. Afkryds den rigtige: (a) (b) (c) Løsning Inddel med endepunkter ∆A = (2 − 1)(2 − 1) = 1 V ≈ f(1,1)∆A = 1 1.17. Integralet som volumen ☞ [S] 12.1 Double integrals over rectangles Eksempel 2 R = [−1,1] × [−2,2], f(x,y) = 1 − x 2 Volumenet under grafen er en halvcylinder med radius 1 og højde 4. Dobbelt integralet findes som volumenet. 1 1 − x2dA = 2 π12 · 4 R = 2π
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CALCULUS "SLIDES" TIL CALCULUS 1 +
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