Final answer:
The mass and center of mass of a solid with non-uniform density δ(x,y,z) are found by setting up and evaluating triple integrals in polar coordinates for the mass and in rectangular coordinates for the center of mass. These integrations take the density function into account and are normalized by the total mass for the center of mass.
Step-by-step explanation:
The student has been asked to find the mass and the center of mass of a solid with given bounds and a non-uniform density function. The first step is to sketch the region of integration. Unfortunately, I cannot provide a graph, but you can visualize the solid by drawing the planes and lines described in the coordinate system. After that, one must set up the integral for the mass using polar coordinates.
To find the mass, we integrate the density function δ(x,y,z) over the volume of the solid. The first step is to set up the double integral in polar coordinates (r, θ) for the projected region onto the xy-plane. Since we're dealing with a three-dimensional solid, we need to integrate over z as well, resulting in a triple integral.
For the center of mass, we use the integrals of the moments about the respective planes divided by the total mass. This involves calculating the first moments of the solid's mass distribution in relation to the x, y, and z-axes. Integrals are calculated with bounds determined from the solid's limits, and the density function δ(x,y,z) is included in the integrations.
The calculation for the center of mass in rectangular coordinates will involve finding the x, y, and z coordinates separately, which are denoted as えₓⱼ, えₓⱽ, and えₓⱾ respectively, and are given by the respective integrals of xρ(x,y,z), yρ(x,y,z), and zρ(x,y,z) over the volume V, normalized by the total mass M:
えₓⱼ = (1/M)∫∫∫ xδ(x,y,z) dV
えₓⱽ = (1/M)∫∫∫ yδ(x,y,z) dV
えₓⱾ = (1/M)∫∫∫ zδ(x,y,z) dV
The actual computation for the mass and the center of mass will need to be done using appropriate calculus techniques, and computational tools like calculators or software may be required to evaluate the final integrals for the center of mass.