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A-region-is-enclosed-by-curves-x-2-4y-x-2-4y-x-4-amp-x-4-V-1-is-the-volume-of-the-solid-obtained-by-rotating-the-above-region-round-the-y-axis-Another-regions-consists-of-points-x-y-satisf




Question Number 144771 by imjagoll last updated on 29/Jun/21
A region is enclosed by curves  x^2 =4y, x^2 =−4y, x=4 & x=−4  V_1 is the volume of the solid obtained  by rotating the above region round  the y−axis.  Another regions  consists of points (x,y) satisfying  x^2 +y^2 ≤16, x^2 +(y−2)^2 ≥4 and  x^2 +(y+2)^2 ≥4 ,V_2  is the volume  of the solid obtained by rotating  this region round the y−axis   Then V_1 =...
$$\mathrm{A}\:\mathrm{region}\:\mathrm{is}\:\mathrm{enclosed}\:\mathrm{by}\:\mathrm{curves} \\ $$$$\mathrm{x}^{\mathrm{2}} =\mathrm{4y},\:\mathrm{x}^{\mathrm{2}} =−\mathrm{4y},\:\mathrm{x}=\mathrm{4}\:\&\:\mathrm{x}=−\mathrm{4} \\ $$$$\mathrm{V}_{\mathrm{1}} \mathrm{is}\:\mathrm{the}\:\mathrm{volume}\:\mathrm{of}\:\mathrm{the}\:\mathrm{solid}\:\mathrm{obtained} \\ $$$$\mathrm{by}\:\mathrm{rotating}\:\mathrm{the}\:\mathrm{above}\:\mathrm{region}\:\mathrm{round} \\ $$$$\mathrm{the}\:\mathrm{y}−\mathrm{axis}.\:\:\mathrm{Another}\:\mathrm{regions} \\ $$$$\mathrm{consists}\:\mathrm{of}\:\mathrm{points}\:\left(\mathrm{x},\mathrm{y}\right)\:\mathrm{satisfying} \\ $$$$\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} \leqslant\mathrm{16},\:\mathrm{x}^{\mathrm{2}} +\left(\mathrm{y}−\mathrm{2}\right)^{\mathrm{2}} \geqslant\mathrm{4}\:\mathrm{and} \\ $$$$\mathrm{x}^{\mathrm{2}} +\left(\mathrm{y}+\mathrm{2}\right)^{\mathrm{2}} \geqslant\mathrm{4}\:,\mathrm{V}_{\mathrm{2}} \:\mathrm{is}\:\mathrm{the}\:\mathrm{volume} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{solid}\:\mathrm{obtained}\:\mathrm{by}\:\mathrm{rotating} \\ $$$$\mathrm{this}\:\mathrm{region}\:\mathrm{round}\:\mathrm{the}\:\mathrm{y}−\mathrm{axis}\: \\ $$$$\mathrm{Then}\:\mathrm{V}_{\mathrm{1}} =… \\ $$$$\: \\ $$

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