Hydrostatic Force on a Plane Surface

[Arc_Unbated]

Homework Statement

I am confused with the derivation of Hydrostatic force on a plane surface. What is confusing me is, how can the integral of the differential force get the resultant force for the entire area of the surface? The differential force is as follows:
dF = ρgh dA

and the magnitude of the resultant force can be obtained by integrating the differential force over the whole area:
dF = ρg∫h dA

Whats bothering me is, should this not be a double integral, to integrate in the x direction, and then the y direction for the entire surface? I don't understand how integrating with respect to dA will add up all the dA's to give the total surface area.

The full derivation is on this page:
https://ecourses.ou.edu/cgi-bin/eBook.cgi?doc=&topic=fl&chap_sec=02.3&page=theory

Basically, why does the integral of ρgh dA give the total area? I can understand that it would sum up the individual dA's on a single line on the surface, but I can't understand how it adds up all the dA's that are not on the same line, but on another position on the plane.

Thanks for your help

 

[LawrenceC]

In the derivation there is a symbol A at the base of the integral sign. That denotes that the integration is over the entire area of the plate which might well be a double integral. It's merely a shorthand notation. dA could also have been written as wdy where w is the width of the plate (z direction). In that case it would be a single integral.

 

[Arc_Unbated]

In the derivation there is a symbol A at the base of the integral sign. That denotes that the integration is over the entire area of the plate which might well be a double integral. It's merely a shorthand notation. dA could also have been written as wdy where w is the width of the plate (z direction). In that case it would be a single integral.

thank you very much that answeres my question. Really appreciate that, i was getting frustrated trying to follow the logic bit by bit through the derivation in my notes, and then I got stuck there and thought i would just have to accept it. Now i understand, thanks again