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?

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    $\begingroup$ You appear to be talking about a path integral. You need to give some physical context to get a physical context. Mathematics is mathematics--you can certainly set up the integral you proposed but it can be difficult to interpret the results (such as interpreting the "area underneath the curve"). And it's very important to realize that an integral does not represent the area underneath a curve--rather the area underneath a curve can be an interpretation of an integral (but the area is not necessarily the correct interpretation). $\endgroup$ – Jared May 26 '16 at 1:13
  • $\begingroup$ When you say $g(x, y): \mathbb{R}^2 \mapsto \mathbb{R}$ you are talking about a 3D surface (a 2D, curved, surface in 3D space). When you say there is a function $y = f(x)$ you are talking about a path on the 2D plane that traverses your 3D surface. Then when you say $\int g(x, f(x))dx$ you are talking about something that makes very little physical sense (but mathematically is perfectly valid). $\endgroup$ – Jared May 26 '16 at 1:25
  • $\begingroup$ @Jared Thanks! Your first comment and reading up on line integral cleared up a lot! $\endgroup$ – Sep Jun 3 '16 at 17:10