Now the proof for the parallel axis theorem is fairly easy to follow but I couldn't understand a part which says,
$\int{dm}{\vec{r_c}}={\vec{0}}$
This sums up the product of each particle's mass and it's perpendicular distance from the axis passing through the centre of mass.
Can somebody please explain how this is true

Now the proof for the parallel axis theorem is fairly easy to follow but I couldn't understand a part which says,
$$\int{dm}{\vec{r_c}}={\vec{0}}.$$
This sums up the product of each particle's mass and it's perpendicular distance from the axis passing through the centre of mass.
Can somebody please explain how this is true