I would like to have some clarification on the physical meaning of $dx$. I already know the following in the context of the area under the curve:
$\lim_{\Delta x \rightarrow 0} \sum f(x) \Delta x \approx \int f(x) dx $
$dx$ is still an interval on x axis. Makes perfect sense.
Let's say I have the following curve $(x,f(x))$ like this:
and I have some function $g(x,y)$ that I want to measure its total sum along my curve. Can I formulate it is as?
$\int_{a}^b g(x,f(x)) dx$
If so, what is the physical meaning of $dx$ here? Aren't we multiplying some extra values ($dx$) into $g(x,f(x))$ and getting a wrong result?