第1个回答 2019-08-31
设直线x+y=π/2,将区域D分为D1,D2,
其中D1={(x,y)|0<=x<=π/2,0<=y<=π/2-x},
D2 = {(x,y)|0<=x<=π/2,π/2-x<=y<=π/2},
则∫∫|cos(x+y)|dxdy = (D1)∫∫cos(x+y)dxdy-(D2)∫∫cos(x+y)dxdy,
(D1)∫∫cos(x+y)dxdy = ∫(0至π/2)dx∫(0至π/2-x)cos(x+y)dy = ∫(0至π/2)(1-sinx)dx = π/2-1
(D2)∫∫cos(x+y)dxdy=∫(0至π/2)dx∫(π/2-x至π/2)cos(x+y)dy = ∫(0至π/2)(1-sinx)dx = 1-π/2,
所以∫∫|cos(x+y)|dxdy = π/2-1-(1-π/2) = π-2。