Calculus – Intuitive Meaning of Line Integral with Respect to x

Question

I know how the line integral is calculated, but I want to know what the meaning of line integral with respect to x is, I mean, intuitively.
The integral is $\int_C f(x,y) dx$ or it can be $\int_C f(x,y) dy$. ($C$ is a curve here)
Thanks.

Answer 1

You can think of it as representing work done moving along the curve C with a force of strength $f$ pointing in the $x$-direction. (So if $x$ is vertical, then f might represent gravity, and the integral would be work done against gravity).

Mathematically, it essentially integrates the dot product of the unit tangent vector of C with a vector (f,0,0), thus measuring how much the curve lines up with or against a vector field of strength f in the x direction.