∫∫∫xyzdxdydz,积分区域:x≥0,0≤y≤1,z≥0,x+z≤1∫∫∫(x+y+z)dxdydz,积分区域：x+y+z≤1,x≥0,y≥0∫∫∫xyzdxdydz,积分区域:x≥0,0≤y≤1,z≥0,x+z≤1∫∫∫(x+y+z)dxdydz,积分区域：x+y+z≤1,x≥0,y≥0,z≥0
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∫∫∫xyzdxdydz,积分区域:x≥0,0≤y≤1,z≥0,x+z≤1∫∫∫(x+y+z)dxdydz,积分区域：x+y+z≤1,x≥0,y≥0∫∫∫xyzdxdydz,积分区域:x≥0,0≤y≤1,z≥0,x+z≤1∫∫∫(x+y+z)dxdydz,积分区域：x+y+z≤1,x≥0,y≥0,z≥0
∫∫∫xyzdxdydz,积分区域:x≥0,0≤y≤1,z≥0,x+z≤1∫∫∫(x+y+z)dxdydz,积分区域：x+y+z≤1,x≥0,y≥0
∫∫∫xyzdxdydz,积分区域:x≥0,0≤y≤1,z≥0,x+z≤1
∫∫∫(x+y+z)dxdydz,积分区域：x+y+z≤1,x≥0,y≥0,z≥0
∫∫∫xyzdxdydz,积分区域:x≥0,0≤y≤1,z≥0,x+z≤1∫∫∫(x+y+z)dxdydz,积分区域：x+y+z≤1,x≥0,y≥0∫∫∫xyzdxdydz,积分区域:x≥0,0≤y≤1,z≥0,x+z≤1∫∫∫(x+y+z)dxdydz,积分区域：x+y+z≤1,x≥0,y≥0,z≥0
1.∫∫∫xyzdxdydz,积分区域:x≥0,0≤y≤1,z≥0,x+z≤1
原式=【0,1】∫ydy【0,1】∫xdx【0,1-x】∫zdz
=【0,1】∫ydy【0,1】(1/2)∫x(1-x)²dx=【0,1】(1/2)∫ydy【0,1】∫(x-2x²+x³)dx
=【0,1】(1/2)∫ydy[x²/2-(2/3)x³+(1/4)x⁴]【0,1】
=【0,1】(1/2)∫ydy[1/2-2/3+1/4]=(1/24)(y²/2)【0,1】=1/48
2.∫∫∫(x+y+z)dxdydz,积分区域：x+y+z≤1,x≥0,y≥0,z≥0
原式=【0,1】∫dx【0,1-x】∫dy【0,1-x-y】∫(x+y+z)dz
=【0,1】∫dx【0,1-x】∫dy[(x+y)z+z²/2]【0,1-x-y】
=【0,1】∫dx【0,1-x】∫[(x+y)(1-x-y)+(1-x-y)²/2]dy
=【0,1】∫dx【0,1-x】(1/2)∫(1-x-y)(1+x+y)dy
=【0,1】(1/2)∫dx【0,1-x】∫(1-x²-2xy-y²)dy
=【0,1】(1/2)∫dx[(1-x²)y-xy²-y³/3]【0,1-x】
=【0,1】(1/2)∫[(1-x²)(1-x)-x(1-x)²-(1-x)³/3]dx
=【0,1】(1/2)∫(1/3)(4x³-3x²-3x+2)dx
=(1/6)[x⁴-x³-(3/2)x²+2x]【0,1】=(1/6)[1-1-(3/2)+2]=1/12.
∫∫∫dxdydz,积分区域：x+y+z≤1,x≥0，y≥0，z≥0
说白了就是求体积
三条棱长都是1，体积=底×高/3=1/3