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(Solved): Why does distance between point and z-axis being propritonate make the density function k(sqrt(x^2 ...



Use spherical coordinates to find the mass of the sphere \( x^{2}+y^{2}+z^{2}=a^{2} \) with the given density.
The density at

Why does distance between point and z-axis being propritonate make the density function k(sqrt(x^2+y^2))?

Use spherical coordinates to find the mass of the sphere \( x^{2}+y^{2}+z^{2}=a^{2} \) with the given density. The density at any point is proportional to the distance of the point from the \( z \)-axis. Step-by-step solution Show all steps [3 \( 75 \% \) (12 ratings) for this solution Step 1/3 Consider the following sphere: \[ x^{2}+y^{2}+z^{2}=a^{2} \] The objective is to find the mass of the sphere, using spherical coordinates. The spherical coordinates are, \[ x=\rho \sin \phi \cos \theta, y=\rho \sin \phi \sin \theta, z=\rho \cos \phi \] The density at any point is proportional to its distance from the z - axis. The density function is, \[ \begin{array}{r} \rho(x, y, z) \alpha \sqrt{x^{2}+y^{2}} \\ \rho(x, y, z)=k\left(\sqrt{x^{2}+y^{2}}\right) \end{array} \]


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Now in general the distance of a 3d point from the z-axis is simply the distance of the p
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