19/02/05 11:36:04.34 xI3EwwZt.net
>>692 (下) 続き
D~ = {(X,Y)| X+Y≦π, 0≦Y≦X } = {(X,Y)| 0≦X≦π, 0≦Y≦min(X,π-X)}
より
∬_D xy sin(π(xx+yy)) dy dx
= (1/2π)^2 (1/2)∬_D~ sin(X+Y) dY dX
= (1/2π)^2 ∫[0,π] ∫[0,min(X,π-X)] sin(X+Y) dY dX
= (1/2π)^2 ∫[0,π/2] ∫[0,X] sin(X+Y) dY dX + (1/2π)^2 ∫[π/2,π] ∫[0,π-X] sin(X+Y) dY dX
= (1/2π)^2 ∫[0,π/2] {cos(X) - cos(2X)}dX + (1/2π)^2 ∫[π/2,π] {cos(X) - cos(π)}dX
= (1/2π)^2 [ sin(X) -(1/2)sin(2X) ](X=0,π/2) + (1/2π)^2 [ sin(X) +X ](X=π/2,π)
= (1/2π)^2 + (1/2π)^2 (π/2 -1)
= (1/2π)^2 (π/2)
= 1/(8π),