Integration for thin spherical shells

Outside

Assuming[r≥R&&R>0,(∫_(r-R)^(r+R))((((r^2)-(R^2))/(s^2)) +1)\[DifferentialD]s]

Inside

Assuming[0<r<R&&R>0,(∫_(R-r)^(R+r))((((r^2)-(R^2))/(s^2)) +1)\[DifferentialD]s]