This question has intriqued me because two methods are generally employed to derive the shape of the surface(paraboloid) .
Both methods give equal numerical answer but differ in which of them P or Q (in figure) has greater pressure
(1) Firstly as done in the book one applies Bernoulli and finds lesser pressure at Q due to it having more velocity.
(2) One can calculate net force (gravity and centrifugal) at a point x distance away from centre and make the surface's tangent perpendicular to it , giving us a solvable differential equation.
The second method predicts more pressure at Q due to it being towards the direction of centrifugal force.
Also if one considers a point R above Q at the surface what will be its pressure.
Logic says , it is equal to pressure at P because both are at surface , but then pressure at Q is greater than at R ( it is below R) so doesnt that mean pressure at Q is greater than at P.
Please explain this situation concluding about exact pressures at P , Q and R and their corresponding reasons.