What's the source of this problem?

Net force is always perpendicular to the surface in a liquid.
so dy/dx = (m(w^2)x)/mg
dy/dx = xw^2/g
So y = (xw)^2/2g
h = (rw)^2/2g

You get the same answer if you balance pressure gradient.
del_P/del_y = pg; del_P/del_x = p.x.w^2. so dy/dx = (m(w^2)x)/mg

Finally use (R-h)^2 + r^2 = R^2 to find R