求二重积分∫∫（x²-y²）dσ,其中D是闭区域：0≤y≤sinx,0≤x≤π.

求二重积分∫∫（x²-y²）dσ,其中D是闭区域：0≤y≤sinx,0≤x≤π.
求二重积分∫∫（x²-y²）dσ,其中D是闭区域：0≤y≤sinx,0≤x≤π.

求二重积分∫∫（x²-y²）dσ,其中D是闭区域：0≤y≤sinx,0≤x≤π.
∫∫（x²-y²）dσ
=∫[0,π]∫[0,sinx]（x²-y²）dydx
=∫[0,π]（x²y-y^3/3）[0,sinx]dx
=∫[0,π]（x^2sinx-sinx^3/3）dx
=∫[0,π]x^2sinxdx-∫[0,π]sinx^3/3dx
=-∫[0,π]x^2dcosx+∫[0,π](1-cos^2 x)/3dcosx
=-x^2cosx[0,π]+2∫[0,π]xcosxdx+(cosx-cos^3x /3)/3[0,π]
=π^2+2∫[0,π]xdsinx-2/9-2/9
=π^2-4/9+2∫[0,π]xdsinx
=π^2-4/9+2xsinx[0,π]-2∫[0,π]sinxdx
=π^2-4/9+2cosx[0,π]
=π^2-4/9-4
=π^2-40/9

∫∫(x^2-y^2)dσ
=∫∫(x^2-y^2)dxdy
=∫[0,π]dx∫[0,sinx](x^2-y^2)dy
=∫[0,π]dx*(y*x^2-y^3/3)|[0,sinx]
=∫[0,π][x^2*sinx-(sinx)^3/3]dx
=∫[0,π]x^2*sinx*dx-1/3*∫[0,π](sinx)^3*dx
=(-x^2*cosx+2xsinx+2cosx)|[0,π]-2/3*∫[0,π/2](sinx)^3*dx
=(π^2-2)-2-2/3*2/3
=π^2-4-4/9
=π^2-40/9

先化为累次积分，我们先固定x求对y的积分有
原积分=∫(x=0->π)x²sinx-1/3(sinx)^3 dx
分部积分可以算出
原积分=π²-40/9