Question

find the total mass of the solid ball x^2+y^2+z^2=1. The ball has a mass density of 1/(1+x^2+y^2+z^2). I know that a conversion to spherical coordinates is needed but am unsure of how to do it.

Answer 1

Let \[\rho^2 = x^2+y^2+z^2\] and triple-integrate the density function over the domain\[\rho \in [0,1]; \theta \in [0,2\pi]; \phi \in [0,\pi]\]
So you should end up with something like\[\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi}\int\limits_{0}^{1}\frac{ 1 }{ 1+\rho^2 }d \rho d \phi d \theta\]