Why Is the Dam Wall Area Constant in Fluid Pressure Calculations?

[Enzo]

Heya folks at Physics forums.

I'm having a bit of a problem with the theory of maths behind a simple formula used in F.M. This is purely a problem dealing with with mathematics rather than fluid dynamics. You'll have to excuse me, I really am a bit slow when it comes to these things, and right now I'm feeling like a complete idiot.


The situation is a basic dam setting. There's a dam wall, and one side of the dam is filled with water. The equation is used in finding the total force created by the water on dam wall.

δF = ρgy*δy

This equation stems from F=PA, where A=y*x, where x=1 in this scenario.
This is later on used with an integration to find the entire force on the dam wall:

(I'm using (S) as the integration sign)

F = ρg (S)y*δy, with the limits being the entire dam wall length in the y direction

What I don't understand is:
Why does the equation small change the length of the wall in respect to the area, rather than to small change the length down the wall in terms of pressure (eg, since pressure changes, why not small change the pressure and keep the dam wall area constant? which would give the equation: δF = ρgδy*y, then integrate that formula over the entire dam wall depth) I know that it would equal the same thing, but still..

Also, since the pressure on the wall changes according to depth, why isn't the depth in the pressure part small changing too? Eg, δF = ρgδy*δy

I apologize if this post is too basic(or stupid), but I'm a bit confused here. Anything you can say that would help me understand would be greatly appreciated.

 

[Dr.D]

It might help if we first write the pressure as a function of the depth, y:

P(y) = rho*g*y

then to get the differential force, you need to multiply this pressure by an element of area,

dF = P(y) * dA = P(y)*1*dy = rho*g*y * dy

This gets your original form that you seemed to be questioning. As to whether you should write this as rho*g*y*dy or rho*g*dy*y, it really makes no difference since the multiplications are commutative.

I hope this same development also explains why it should not by rho*g*dy*dy; that would be completely wrong.