作业帮三重积分比较I1,I2大小设Ω由平面x+y+z+1=0,x+y+z+2=0,x=0,y=0,z=0围成,I1=∫∫∫[ln(x+y+z+3)]²dV,I2=∫∫∫(x+y+z)²dV,比较I1,I2大小
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![三重积分比较I1,I2大小设Ω由平面x+y+z+1=0,x+y+z+2=0,x=0,y=0,z=0围成,I1=∫∫∫[ln(x+y+z+3)]²dV,I2=∫∫∫(x+y+z)²dV,比较I1,I2大小](/uploads/image/z/9283675-67-5.jpg?t=%E4%B8%89%E9%87%8D%E7%A7%AF%E5%88%86%E6%AF%94%E8%BE%83I1%2CI2%E5%A4%A7%E5%B0%8F%E8%AE%BE%CE%A9%E7%94%B1%E5%B9%B3%E9%9D%A2x%2By%2Bz%2B1%3D0%2Cx%2By%2Bz%2B2%3D0%2Cx%3D0%2Cy%3D0%2Cz%3D0%E5%9B%B4%E6%88%90%2CI1%3D%E2%88%AB%E2%88%AB%E2%88%AB%5Bln%28x%2By%2Bz%2B3%29%5D%26%23178%3BdV%2CI2%3D%E2%88%AB%E2%88%AB%E2%88%AB%28x%2By%2Bz%29%26%23178%3BdV%2C%E6%AF%94%E8%BE%83I1%2CI2%E5%A4%A7%E5%B0%8F)
三重积分比较I1,I2大小设Ω由平面x+y+z+1=0,x+y+z+2=0,x=0,y=0,z=0围成,I1=∫∫∫[ln(x+y+z+3)]²dV,I2=∫∫∫(x+y+z)²dV,比较I1,I2大小
三重积分比较I1,I2大小
设Ω由平面x+y+z+1=0,x+y+z+2=0,x=0,y=0,z=0围成,I1=∫∫∫[ln(x+y+z+3)]²dV,I2=∫∫∫(x+y+z)²dV,比较I1,I2大小
三重积分比较I1,I2大小设Ω由平面x+y+z+1=0,x+y+z+2=0,x=0,y=0,z=0围成,I1=∫∫∫[ln(x+y+z+3)]²dV,I2=∫∫∫(x+y+z)²dV,比较I1,I2大小
设Ω由平面x+y+z+1=0,x+y+z+2=0,x=0,y=0,z=0围成,L₁=∫∫∫[ln(x+y+z+3)]²dV；L₂=∫∫∫(x+y+z)²dV,
比较L₁,L₂的大小.
上下限的书写很麻烦,我在积分符号前面用加黑的中括号【a,b】表示,前面的a是
下限,后面的b是上限.
L₁=【Ω】∫∫∫[ln(x+y+z+3)]²dV=【Ω】2∫∫∫ln(x+y+z+3)dxdydz
=2【1,2】∫dx【1-x,2-x】∫dy【1-x-y,2-x-y】∫ln(x+y+z)dz
=2【1,2】∫dx【1-x,2-x】∫dy【1-x-y,2-x-y】∫ln(x+y+z)d(x+y+z)
=2【1,2】∫dx【1-x,2-x】∫{(x+y+z)[ln(x+y+z)-1]}【1-x-y,2-x-y】dy
=2【1,2】∫dx【1-x,2-x】∫[2(ln2-1)+1]dy
=2【1,2】∫dx【1-x,2-x】∫(2ln2-1)dy
=2(2ln2-1)【1,2】∫dx【1-x,2-x】∫dy
=2(2ln2-1)【1,2】∫dx=2(2ln2-1)=0.7726
L₂=【Ω】∫∫∫(x+y+z)²dV=【Ω】∫∫∫(x+y+z)²dxdydz
=【1,2】∫dx【1-x,2-x】∫dy【1-x-y,2-x-y】∫(x+y+z)²dz
=【1,2】∫dx【1-x,2-x】∫dy【1-x-y,2-x-y】∫(x+y+z)²d(x+y+z)
=【1,2】∫dx【1-x,2-x】∫[(x+y+z)³/3]【1-x-y,2-x-y】dy
=【1,2】∫dx【1-x,2-x】∫[8/3-1/3)dy
=(7/3)【1,2】∫dx【1-x,2-x】∫dy
=(7/3)【1,2】∫dx=7/3=2.3333
故L₂>L₁.