Ok so what I want to know is, is this valid? If so what does it mean?
The notation is unusual. It might simply mean F(0)dx.
The integral comes from finding the work done by a force over a distance dx. The force may or may not be variable so I needed to prove the variableness of F did not matter over an infinitesimal for the purposes of determining the amount of work done.
So dw = F⋅dx => F0⋅dx
So integrating over the distance dx would prove this.
I'm doing a proof that I'd like to make rigorous. This makes sense to me visually - we are simply looking at the area of the first strip of dx - which would be the initial value of F. However I cannot find a way to prove it mathematically.
You first need to define an integral with an upper limit dx.
An upper limit of dx would be the area of the first strip of dx. I need to prove the above.
Doh. The area of that first strip would be the height (F0) by the width (dx.) Ok problem solved.
Notation quibble: dx is used for the differential. If you are trying to describe a small non-zero width, use Δx.
It has to be along a distance dx.
Like mathman said, use Δx. dx has its own meaning. Δx can represent some small distance along the x-axis.
dx is the width of each strip. dx would then be the width of the first strip. Δx is the width of a finite number of strips, which is not what I am looking for.
Thirding what the others said: the width of the first strip is ##\Delta x_0##, the width of the second strip is ##\Delta x_1## etc. The width of a finite number of strips is ##\Sigma \Delta x_n##.
dx is not used for the width of a strip.
Separate names with a comma.