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Question Number 144528 by imjagoll last updated on 26/Jun/21

$$\mathrm{Find}\mathrm{the}\mathrm{volume}\mathrm{of}\mathrm{the}\mathrm{region}$$$$\mathrm{bounded}\mathrm{by}\mathrm{the}\mathrm{elliptic}\mathrm{paraboloid}$$$$\mathrm{z}=4-{\mathrm{x}}^2-\frac{1}{4}{\mathrm{y}}^2\mathrm{and}\mathrm{the}\mathrm{plane}\mathrm{z}=0$$$$$$

Answered by EDWIN88 last updated on 03/Jul/21

$$\mathrm{vol}=4{\int }_0^2{\int }_0^{2\sqrt{4-{\mathrm{x}}^2}}(4-{\mathrm{x}}^2-\frac{1}{4}{\mathrm{y}}^2)\mathrm{dy}\mathrm{dx}$$$$=4{\int }_0^2{(4\mathrm{y}-{\mathrm{x}}^2\mathrm{y}-\frac{1}{4}\frac{{\mathrm{y}}^3}{3})}_0^{2\sqrt{4-{\mathrm{x}}^2}}\mathrm{dx}$$$$=16\pi$$

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