Question Number 213661 by Spillover last updated on 13/Nov/24
Commented by MathematicalUser2357 last updated on 14/Nov/24
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Answered by mahdipoor last updated on 14/Nov/24
$${M}\overset{−} {{y}}=\int_{{M}} {ydm}\:\Rightarrow\rho={cte}\:,\:{dm}=\rho{dV} \\ $$$${V}\overset{−} {{y}}=\int_{{V}} {ydV}=\int_{\mathrm{0}} ^{\:\mathrm{1}} {y}\left(\pi{z}^{\mathrm{2}} {dy}\right)=\pi\int_{\mathrm{0}} ^{\:\mathrm{1}} {y}\left(\frac{{y}}{\mathrm{4}}\right){dy}=\frac{\pi}{\mathrm{12}} \\ $$$${V}=\int_{\mathrm{0}} ^{\:\mathrm{1}} \left(\pi{z}^{\mathrm{2}} \right){dy}=\frac{\pi}{\mathrm{8}} \\ $$$$\Rightarrow\overset{−} {{y}}=\:\frac{\pi/\mathrm{12}}{\pi/\mathrm{8}}=\frac{\mathrm{2}}{\mathrm{3}} \\ $$$$ \\ $$