Is it possible to calculate the volume of a parabolic arch?

Question

Given that you know the equation of a parabola that only has positive values, is it possible to find the volume of the parabolic arch itself? NOT the volume of space underneath the arch. I asking about this for my math exploration because I am trying to find out if I can find a connection between energy efficient/water saving water fountains (the ones you find in a park) and the volume of water it uses. If I can find the volume of the parabolic arch, I am assuming than I can calculate how much water is being sprayed/min/hour etc and continue my exploration from there?

EDIT: is it also possible to calculate the volume of a parabolic arch of water when the water kinda just splatters at the end? Instead of the water being a smooth jet the whole way, it looses pressure and just sprinkles out?

Answer 1

You need the thickness of the arch as a function of $x$. You have two functions, one for the top of the arch, call it $y_1(x)$ and one for the bottom, call it $y_2(x)$ The area of the arch is then $\int(y_1(x)-y_2(x))dx$ and (if the thickness is constant) you multiply by the thickness and have the volume. Otherwise, let the thickness be $t(x)$ and the volume becomes $\int(y_1(x)-y_2(x))t(x)dx$ If the thickness varies in $y$ as well you need a double integal.

Answer 2

If you are measuring amount of water flowing per hour then it is not enough to measure an area in a plane parallel to the flow. The flow depends also on the width of the stream perpendicular to that plane, on the cross-sectional shape, and on the velocity of the water (which determines how frequently the water in the air at any moment is exchanged for new water).

I think it is simpler to take the cross-sectional area of the outlet where the water first comes out (which is easier to measure accurately than the flowing part near the top of the arc anyway), and multiply by the velocity of the water passing through that point. This is a standard way to measure a rate of flow.

You can estimate the velocity by applying the equations of motion of a projectile to the path of the water. It should be enough to measure the angle of the jet from horizontal at the outlet, and the height of the top of the arc above the outlet.

There are sources of error involved in this: viscosity, air resistance, errors of measurement, and so forth, but I think probably less severe errors than those you would get by using the method suggested in the question.