center of mass (doubt)

using the method of integration for bodies having continuous distribution of mass.... we use...

Xcm = 1/M ∫x.dm

we all know this (not sure)...

but here's what is annoying me...

on finding the COM of a ring...
we set y = r sinθ... fine....

but in case of a semi-circular disc...
we consider small rings and set
y = 2r/pi

now what's this....????
why not r.sinθ again or something like that....

please tell me how can we take y = 2r/pi here...
??

and how do we decide r in different cases...!!