计算积分∫∫Dxycos(xy^2)dxdy=?其中D:0≤x≤π/2,0≤y≤2∫∫的下面是D

Question

计算积分∫∫Dxycos(xy^2)dxdy=?其中D:0≤x≤π/2,0≤y≤2 ∫∫的下面是D

Answer 1

I = ∫∫D x y cos(x y²) dxdy 先 y 后 x
= ∫ [0,π/2] dx ∫ [0,2] x y cos(x y²) dy
= ∫ [0,π/2] (1/2) sin(x y²) | [0,2] dx
= (1/2) ∫ [0,π/2] sin(4x) dx
= (-1/8 cos(4x) | [0,π/2]
= 0