MvCalc51

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The volume is given by V=\iint _{R}\int _{{x^{2}+y}}^{{4}}dzdydx\, where R is the region in the (x,y)plane bounded by the circle x^{2}+y^{2}=4\,,we therefore have

V=\iint _{R}[4-(x^{2}+y^{2})]dydx\,

=4\int _{{0}}^{{{\frac  {\pi }{2}}}}\int _{0}^{2}(4-r^{2})rdrd\theta \,

=4{\frac  {\pi }{2}}[4{\frac  {r^{2}}{2}}-{\frac  {r^{4}}{4}}]_{0}^{2}\,

=2\pi [2r^{2}-{\frac  {r^{4}}{4}}]_{0}^{2}\,

=2\pi (8-4)=8\pi \,

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