求函数(2x-y)的二重积分,D是由y=1,2x-y+3=0,x+y-3=0围成的区域.
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积分范围是： 1<=y<=3,3-y<=x<=(y-3)/2
∫dy∫(2x-y)dx
=∫(x^2-xy)dx,3-y<=x<=(y-3)/2
=∫[(3-y)^2-y(3-y)-(1/4)(y-3)^2+y(y-3)/2]dy
=∫(9y^2/4-9y+27/4)dy,1<=y<=3
=3y^3/4-9y^2/2+27y/4,1<=y<=3
=(3/4)*26-(9/2)*8+27/2
=39/2-36+27/2
=33-36
=-3
∫dy∫(2x-y)dx
=∫(x^2-xy)dx,3-y<=x<=(y-3)/2
=∫[(3-y)^2-y(3-y)-(1/4)(y-3)^2+y(y-3)/2]dy
=∫(9y^2/4-9y+27/4)dy,1<=y<=3
=3y^3/4-9y^2/2+27y/4,1<=y<=3
=(3/4)*26-(9/2)*8+27/2
=39/2-36+27/2
=33-36
=-3
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