Find the center of mass of the solid bounded by z=4 - x² - y² and z=0 with density ρ(x,y,z) = z.
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Step-by-Step Solution
Step 1: Identify the region and density function
The solid is bounded by the paraboloid z=4−x2−y2 and the xy -plane (z=0). The density function is given as ρ(x,y,z)=z. This means the density varies with height.
Step 2: Determine the limits of integration
The intersection of z=4−x2−y2 and z=0 gives 4−x2−y2=0, or x2+y2=4. This is a circle of radius 2 in the xy -plane. Converting to cylindrical coordinates, x2+y2=r2, so the region in the xy -plane is 0≤r≤2 and 0≤θ≤2π. The z limits are from 0 to 4−r2.
Step 3: Calculate the total mass (M)
The total mass M is found by integrating the density function over the volume of the solid. We use cylindrical coordinates, so dV=rdzdrdθ. We integrate with respect to z first, then r, and finally θ. After evaluating the integrals, we find the total mass M=364π.
Step 4: Calculate the moments Mx, My, and Mz
Due to the symmetry of the solid and the density function with respect to the z -axis, the center of mass will lie on the z -axis. This means Mx=0 and My=0, so xˉ=0 and yˉ=0. We only need to calculate Mz.
Step 5: Calculate Mz
We integrate z2⋅r with respect to z, then r, and finally θ. The result of this integration is Mz=364π.
Step 6: Calculate the center of mass (xˉ,yˉ,zˉ)
The coordinates of the center of mass are given by xˉ=MMx, yˉ=MMy, and zˉ=MMz. Since Mx=0 and My=0, we have xˉ=0 and yˉ=0. Substituting the values for Mz and M, we find zˉ=1.