Re: [積分] 請問一題積分問題

[LuisSantos ( )]

※ 引述《flygey (努力達成目標)》之銘言：
:             2  2
: 1.Find∫∫(x +y )dA,where P is the parallelogram bounded by the lines
:   x+y=1, x+y=2, 3x+4y=5, and 3x+4y=6
: 請問一下這題如何解
: 感謝


令 u = x + y     ------(1)

   v = 3x + 4y   ------(2)


(2) - (1)*3 => y = -3u + v 代入(1)得 x = u - y = 4u - v


    | δx     δx |
    |-----   -----|
    | δu     δv |   | 4  -1|
J = |             | = |      | = 4 - 3 = 1  =>  |J| = 1
    | δy     δy |   |-3   1|
    |-----   -----|
    | δu     δv |

1 ≦ x + y ≦ 2  =>  1 ≦ u ≦ 2

5 ≦ 3x + 4y ≦ 6  =>  5 ≦ v ≦ 6


  ∫∫x^2 + y^2 dA

   2  6
= ∫ ∫ ((4u - v)^2 + (-3u + v)^2)*(|J|) dvdu
   1  5

   2  6
= ∫ ∫ (25)(u^2) - 14uv + (2)(v^2) dvdu
   1  5

   2                                2        |v = 6
= ∫ (25)(u^2)(v) - (7)(u)(v^2) + (---)(v^3) |       du
   1                                3        |v = 5

   2                    182
= ∫ (25)(u^2) - 77u + ----- du
   1                     3

    25            77            182      |2
= (----)(u^3) - (----)(u^2) + (-----)(u) |
     3             2             3       |1

   175     231     182
= ----- - ----- + -----
    3       2       3

   357     231               231         7
= ----- - -----   =   119 - -----   =   ---
    3       2                 2          2





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