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A density function captures how matter is dispersed within a given space. In the context of our gaseous star, the density at any point within the star is provided by the function \( \delta = k\left(1- \frac{\rho^2}{a^2}\right) \). Here, \( k \) is a constant representing the maximum density at the center of the star, while \( \rho \) indicates the distance from the center. As you move from the center to the boundary of the star, the density decreases from \( k \) to zero. This spatial variation is vital because it affects how we calculate the total mass of the star. Understanding this function helps you see why integration is necessary: we need to account for all the tiny changes in density as we move outward through the star's volume.

The volume integral is a powerful tool for calculating quantities like mass over a region with varying density. Think of the star as a series of thin, concentric shells. At each radius \( \rho \), the shell has a small volume \( dV = 4\pi \rho^2 d\rho \). By integrating over these shells, we can find the total mass. The formula \[ M = \int_0^a k\left(1- \frac{\rho^2}{a^2}\right) 4\pi \rho^2 d\rho \] illustrates this. The integral collects the mass of each infinitesimal shell from the center of the star (\( \rho = 0 \)) to its outer edge (\( \rho = a \)). Calculating this integral involves splitting it into parts, each easier to solve, simplifying the task of finding the star's total mass.

Calculating the mass of a star with varying density requires integrating its density function over its volume. In our solution, the density function is combined with the shell's volume, yielding an expression for a shell's mass \( dm = \delta \cdot dV \). The final mass integral splits into two simpler parts:

• The first, \( \int_0^a \rho^2 d\rho \), gives \( \frac{a^3}{3} \).
• The second, \( \int_0^a \rho^4 d\rho \), results in \( \frac{a^5}{5} \).

Combining these results shows the total mass of the star as \[ M = 4\pi k \cdot \frac{2a^3}{15} \].This process underscores the importance of integrating over the domain to account for changes in density, which allows you to find the mass of bodies with non-uniform distribution of matter.

Spherical coordinates are ideal for problems involving spheres, like our gaseous star. They use three parameters: radius \( \rho \), polar angle \( \theta \), and azimuthal angle \( \phi \). For our mass calculation, only the radial part \( \rho \) was necessary since the density varies radially, but not angularly. The shell's volume element \( dV = 4\pi \rho^2 d\rho \)is derived from the full spherical volume element, demonstrating why spherical coordinates simplify calculations of radially symmetric bodies. This reduction allows us to focus solely on integrating along the radius, making spherical coordinates a powerful choice for these types of integral problems.