微積分 重積分求解

R是在(X,Y)平面上四邊形, 由A(1,0),B(0,1),C(4,2),D(3,3)所組成Line AB: y/(x-1) = -1 ==> y = -x + 1, dy = -dxLine BC: y/(x-1) = 2/3 ==> y = 2(x - 1)/3, dy = 2dx/3Line CD: (y-3)/(x-3) = -1 ==> y = -x + 6, dy = -dxLine DA: (y-1)x = 2/3 ==> y = (3 + 2x)/3, dy = 2dx/3
∫∫R (3X-2Y)dA
= ∫∫(0~1)(3x+2x-2)dx(-dx) + ∫∫(1~4)(3x-4x/3+4/3)dx(2dx/3)+ ∫∫(4~3)(3x+2x-12)dx(-dx) + ∫∫(3~0)(3x-2-4x/3)dx(2dx/3)
= -∫∫(0~1)(5x-2)(dx)^2 + ∫∫(1~4)(2/9)*(5x+4)(dx)^2+ ∫∫(3~4)(5x-12)(dx)^2 - ∫∫(0~3)(2/9)*(5x-6)(dx)^2= (0~1)(-5x^3/6 + x^2) + (1~4)(2/9)*(5x^3/6 + 2x^2)+ (3~4)(5x^3/6 - 6x^2) - (0~3)(2/9)*(5x^3/6 - 3x^2)
= (-5/6 + 1) + (2/9)*(5*64/6 + 32 - 5/6 - 2) + (5*64/6 - 96 - 5*27/6 + 54) - (2/9)*(5*27/6 - 27)
= 1/6 + (2/9)*(495/6) + 67/6 + (2/9)*27/6
= 34/3 + 55/3 + 3/3
= 92/3
= Answer

四點所圍面積=5
四邊形之形心坐標=(2, 1.5)
故所求積分= 5*(3*2- 2*1.5)= 15

2015-06-24 20:22:04 補充：
雙重積分,與沿邊界的線積分不相同
哪來 dy=-dx, dy= (2/3) dx ?

四邊形的4個邊:
x+y = 1, x+y = 6,
y = (2/3)x + 1, y = (2/3)(x-1)

兩個方法:

一是就原式將積分範圍依 x 分成3段: 0-1, 1-3, 3-4.

二是做雙變數之變數變換:

u = x+y, v = y-(2/3) x

u一v 之積分區域是矩形區域:
u: 1-6
v: -2/3 - 1