Final answer:
To find the mass of a solid in the shape of a sphere with a radius of 8, we need to integrate the mass density function over the volume of the sphere. The mass density is proportional to the square of the distance from the center. Using the spherical coordinate system for integration, we can find the mass in terms of the proportionality constant.
Step-by-step explanation:
To find the mass of a solid in the shape of a sphere with a radius of 8, we need to integrate the mass density function over the volume of the sphere. In this case, the mass density is proportional to the square of the distance from the center. The mass density function can be written as μ = k r^2, where μ is the mass density, k is the proportionality constant, and r is the distance from the center.
The volume of a sphere can be calculated using the formula V = (4/3)πr^3, where V is the volume and r is the radius. We can rewrite the mass density function as μ = k (4/3)πr^2.
Integrating the mass density function over the volume of the sphere, we get the total mass. The mass m can be calculated as m = ∫μdV, where dV is an element of volume. Using the spherical coordinate system for integration, the integral becomes m = ∫μ r^2 sin(θ)dr dθ dφ.
Substituting the mass density function and the limits of integration, we can solve the integral to find the mass of the solid sphere in terms of the proportionality constant. The final solution will be in terms of the proportionality constant k.