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set-x-y-z-x-2-y-2-z-2-1-certificate-4-3-2-1-3-x-2y-2z-5-1-3-dv-8-3-




Question Number 211800 by MrGaster last updated on 21/Sep/24
  set 𝛀={(x,y,z)∣x^2 +y^2 +z^2 ≤1},                       certificate:    ((4𝛑)/3)(2)^(1/3) ≤∫∫_𝛀 ∫((x+2y−2z+5))^(1/3) dv≤((8𝛑)/3)
$$ \\ $$$$\boldsymbol{{set}}\:\boldsymbol{\Omega}=\left\{\left(\boldsymbol{{x}},\boldsymbol{{y}},\boldsymbol{{z}}\right)\mid\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{y}}^{\mathrm{2}} +\boldsymbol{{z}}^{\mathrm{2}} \leq\mathrm{1}\right\}, \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{certificat}}\mathrm{e}: \\ $$$$\:\:\frac{\mathrm{4}\boldsymbol{\pi}}{\mathrm{3}}\sqrt[{\mathrm{3}}]{\mathrm{2}}\leq\int\underset{\boldsymbol{\Omega}} {\int}\int\sqrt[{\mathrm{3}}]{\boldsymbol{{x}}+\mathrm{2}\boldsymbol{{y}}−\mathrm{2}\boldsymbol{{z}}+\mathrm{5}}\boldsymbol{{dv}}\leq\frac{\mathrm{8}\boldsymbol{\pi}}{\mathrm{3}} \\ $$$$ \\ $$$$ \\ $$

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