# Math Help - dense ball

1. ## dense ball

A solid spherical ball of radius $R$ is created in a way such that the density at the point $(x,y,z)$ is proportional to the point's distance from the origin.
What is the mass of the ball?

2. I hope you didn't see my first solution. This time I have actually done some thinking.
The density $\delta(r) = kr$ for some $0 < k$

The surface of a sphere with radius $r$ is $r = 4\pi r^2$
A sphercial shell $dV = 4\pi r^2\,dr$
$dm = \delta\,dV$
$dm = 4k\pi r^3\,dr$
Thus,
$m = 4k\pi\int_0^R r^3\,dr = \pi kR^4$

3. Originally Posted by putnam120
A solid spherical ball of radius $R$ is created in a way such that the density at the point $(x,y,z)$ is proportional to the point's distance from the origin.
What is the mass of the ball?
The rule is,
$m=\int \int_S \int \rho(x,y,z) dV$
In this case,
$\rho(x,y,z)=\kappa \sqrt{x^2+y^2+z^2}=\kappa \rho$

Use spherical coordinates,
$\int_0^{2\pi} \int_0^{\pi} \int_0^R \kappa \rho^3 \sin \phi d\rho\, d\phi\, d\theta$