Problem 8 [15 points] Applications of triple integrals in mechanics: Let p = p(x, y, z) be the density function of t a s

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Problem 8 15 Points Applications Of Triple Integrals In Mechanics Let P P X Y Z Be The Density Function Of T A S 1 (67.88 KiB) Viewed 29 times

Problem 8 [15 points] Applications of triple integrals in mechanics: Let p = p(x, y, z) be the density function of t a solid V. Summing the elements of mass up dm = pdV = pdx dy dz Myz = ₁₁ XpdV, and the coordinates (§, 7, C) of the center of mass =fff, pdx dy dz Using elementary moments d Myz = x dm = xpdV, d Mzx = y dm = ypdV, d Mry = z dm = zp dV, we find the moments SSSv xp dv V m = M₂x - • [[S₁₂ UpdV, n = -Jffy Mp V Yp V M₂y = [[[₁, ²+ p= ZpdV - ffy ZpdV V A. The region V lies between the paraboloiid z = 24 - x² - y² and the cone z = 2√² + 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 k x² + y² + z²