MvCalc55

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The two surfaces intersect on the elliptic cylinder x^{2}+9y^{2}=z=18-x^{2}-9y^{2}\, that is x^{2}+9y^{2}=9\,

The projection of this volume onto xy-plane region D enclosed by ellipse having the same question {\frac  {x^{2}}{3^{2}}}+{\frac  {y^{2}}{1^{2}}}=1^{2}\,

The volume can be covered as

V=\int _{{-3}}^{{3}}\int _{{-{\sqrt  {{\frac  {9-x^{2}}{9}}}}}}^{{{\sqrt  {{\frac  {9-x^{2}}{9}}}}}}\int _{{x^{2}+9y^{2}}}^{{18-x^{2}-9y^{2}}}dzdydx\,

=\int _{{-3}}^{{3}}\int _{{-{\sqrt  {{\frac  {9-x^{2}}{9}}}}}}^{{{\sqrt  {{\frac  {9-x^{2}}{9}}}}}}[18-x^{2}-9y^{2}-x^{2}-9y^{2}]dydx\,

=2\int _{{-3}}^{{3}}\int _{{-{\sqrt  {{\frac  {9-x^{2}}{9}}}}}}^{{{\sqrt  {{\frac  {9-x^{2}}{9}}}}}}(9-x^{2}-9y^{2})dydx\,

=\int _{{-3}}^{{3}}[9y-x^{2}y-3y^{3}]_{{-{\sqrt  {{\frac  {9-x^{2}}{9}}}}}}^{{{\sqrt  {{\frac  {9-x^{2}}{9}}}}}}dx\,

={\frac  {8}{9}}\int _{{-3}}^{{3}}(9-x^{2})^{{{\frac  {3}{2}}}}dx\,

=72\int _{{0}}^{{\pi }}\sin ^{4}\theta d\theta \,

=72\int _{{0}}^{{\pi }}[{\frac  {1-\cos 2\theta }{2}}]^{2}d\theta =27\pi \,

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