Applications of triple integrals in mechanics: Let ρ=ρ(x,y,z) be the density function of t a solid V. Summing the elemen

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Applications of triple integrals in mechanics: Let ρ=ρ(x,y,z) be the density function of t a solid V. Summing the elements of mass up dm=ρdV=ρdxdydz m=∭V​ρdxdydz Using elementary moments dMyz​=xdm=xρ​dV,dMzx​=ydm=yρ​dV,dMxy​=zdm=zρ​dV, we find the moments Myz​=∭V​xρ​dV,Mzx​=∭V​yρ​dV,Mxy​=∭V​zρ​dV and the coordinates (ξ,η,ζ) of the center of mass ξ=V∭V​xρ​dV​,η=V∭V​yρ​dV​,ζ=V∭V​zρ​dV​ A. The region V lies between the paraboloiid z=24−x2−y2 and the cone z=2x2+y2​. Find the centroid of V - the center of mass in the case if the density is constant. B. Find the center of mass of the solid bounded by the surfaces x2+y2=2az,x2+y2+z2=3a2 C. Find mass and the coordinates of the center of mass of the sphere x2+y2+z2≤2az if ρ=x2+y2+z2​k​