What we want to do here is show that the surface of a fluid in a container spinning around a vertical axis at a constant angular speed
For the purpose of analysis, let us imagine isolating a very thin vertical slab of fluid with thickness c, as in the figure below. The central plane of the slab contains the spin axis, and it will suffice to consider the part of the slab which lies to one side of the spin axis.
is a parabolid of revolution whose axis of symmetry is the spin axis. (We assume that the fluid has come to rest with respect to the container.)
Our job is to prove that the profile y = f(x) of the fluid's surface, as indicated in the figure, is a parabola. To do this, consider the small, wedge-shaped block of fluid shaded in light blue. We are assuming that f is differentiable, so if the width 
x of the block is very small, its shape is nearly triangular and its mass is approximately given by
)(c)(xy / 2),
where is the density of the fluid. Since the fluid has come to rest with respect to the rotating container, the little block is moving in a circular orbit about the spin axis with angular speed . If the block is really small, then each part of the block is about x units from the spin axis (the block was deliberately made too large in the figure so you could see it easily), so the basic physics of circular motion says that the block must be held in its orbit by an "inward" force (toward the spin axis) of (mass) x 
. Thus
= c (xy / 2) x .
But what provides this force? The answer is the fluid pressure of the surrounding fluid. The outer face of the block (away from the spin axis - to the right in the figure) is a narrow rectangle of width c and height y. Now recall one of the first applications of integration you ever saw, namely, computing the total force that a fluid exerts on a dam. Using that method, we can determine that the total fluid force on the outer face is given by
where g is the acceleration of gravity. Equating the fluid force with the inward force, we get
g c (y) / 2 = c (xy / 2) x .
Multiplying both sides of this equation by 2 / (g c xy) gives
y /x = ( / g) x.
This equation becomes more accurate as x 
0, and taking the limit gives
/ g) x.Integrating this equation and noticing in our diagram that f(0) = 0, we see that
/ 2g) x,so the surface profile is indeed a parabola! Moreover, this equation is independent of the direction in which we choose the slab (as long as it contains the spin axis), so all profiles are the same, and the surface is a paraboloid of revolution. We get more than this, however. We can write the last equation as
/ 2g) x = 4(g / 2) y.
Recalling that x = 4Ly is the equation of a parabola with focal length L (the distance from the vertex to the focus), we see that the focal length L of the paraboloid is
).This is of course a very important fact in the design of liquid mirror telescopes. According to a liquid mirror web page at the Université Laval in Québec, "For large mirrors of practical interest the periods of rotation are of the order of 10 seconds." Suppose that a liquid mirror is rotating, say, once every 5 seconds. What is the focal length (in feet) of such a mirror? You can check the solution or go back to Liquid Mirror Telescopes.