For my own knowledge and to understand why. I am trying to convince myself that the center of mass for a rigid solid sphere is at the origin (0,0,0).
I begin with the basic definition of CM
$$\overrightarrow{r}_{cm}=\frac{\sum_i m_ir_i}{\sum_i m_i}$$
Turning this into an integral form where the body is continuous:
$$\overrightarrow{r}_{cm}=\frac{1}{M}\int r\space dm$$
To convert dm into something we can integrate:
I know that $dm=\rho \space dV$
I know the the volume of a sphere is: $V=\frac{4}{3} \pi r^3$
So $dV=4\pi r^2 \space dr$
Ultimately giving me an integral that I can integrate over the the entire slice of the sphere where the origin is at 0,0,0:
$$r_{cm}= \frac{4\pi \rho}{m}\int_{-r}^{r} r^3 \space dr$$
If you integrate this and try to solve it you find that $r_{cm}=0$ which is exactly what I want, but I would like to know if this is the proof that I need to convince myself that the center of mass on solid sphere is located at the center of the structure or did i stumble upon this by coincidence and I made a mistake somewhere along the line and just got lucky to get to the answer?