MvCalc53

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V=\int _{0}^{1}\int _{{0}}^{{1-x}}\int _{{0}}^{{1-x^{2}-y^{2}}}dzdydx\,

=\int _{0}^{1}\int _{{0}}^{{1-x}}(1-x^{2}-y^{2})dydx\,

=\int _{0}^{1}[y-x^{2}y-{\frac  {y^{3}}{3}}]_{{0}}^{{1-x}}dx\,

=\int _{0}^{1}[1-x-x^{2}(1-x)-{\frac  {(1-x)^{3}}{3}}dx\,

=[x-{\frac  {x^{2}}{2}}-{\frac  {x^{3}}{3}}+{\frac  {x^{4}}{4}}-{\frac  {(1-x)^{4}}{-12}}]_{0}^{1}\,

=1-{\frac  {1}{2}}-{\frac  {1}{3}}+{\frac  {1}{4}}+{\frac  {1}{12}}[(1-x)^{4}]_{0}^{1}\,

=1-{\frac  {1}{2}}-{\frac  {1}{3}}+{\frac  {1}{4}}-{\frac  {1}{12}}\,

={\frac  {1}{3}}\,

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