求二重积分 ∫∫Dsin(x+y)dσ,其中D={(x,y)|0≤x≤π/2,0≤y≤π/2};

求二重积分
∫∫_Dsin(x+y)dσ,其中D={(x,y)|0≤x≤π/2,0≤y≤π/2};
【正确答案】：解：∫∫_Dsin(x+y)dσ=∫∫_Dsin(x+y)dxdy =∫^π/2₀dx∫^π/2₀sin(x+y)dy =∫^π/2₀[-cos(x+y)]|^π/2₀dx =∫^π/2₀[cos-cos(x+π/2)]dx =∫^π/2₀(cosx+sinx)dx =(sinx-cosx)|^π/2₀=2