Question

高等数学 计算二重积分 要求有步骤

Answer 1

例67 I = ∫∫<D> xyf''<xy>(x,y)dxdy
= ∫<0, 1>xdx∫<0, 1> yf''<xy>(x,y)dy
= ∫<0, 1>xdx∫<0, 1> ydf'<x>(x,y)
= ∫<0, 1>xdx{[yf'<x>(x,y)]<0,1>-∫<0, 1>f'<x>(x,y)dy}
= ∫<0, 1>xf'<x>(x,1)dx -∫<0, 1>xdx∫<0, 1>f'<x>(x,y)dy
= ∫<0, 1>xdf(x,1)-∫<0, 1>dy∫<0, 1>xf'<x>(x,y)dx
= [xf(x, 1)]<0, 1>-∫<0, 1>f(x,1)dx-∫<0, 1>dy∫<0, 1>xdf(x,y)
= 0 -∫<0, 1>dy{[xf(x,y)]<0, 1> -∫<0, 1>f(x,y)dx}
= -∫<0, 1>dy{0-∫<0, 1>f(x,y)dx}
= ∫<0, 1>dy∫<0, 1>f(x,y)dx = a