Line Integral Interpretation.

Question

I am trying to brush up on Calc III material but I am at wall when it comes to understanding what a line integral means. I understand it is the 2D area of a sheet that goes from the curve $C$ to the function value above the curve. Or if $f(x,y)$ represents a mass density function, then the line integral represents the mass of the curve $C$. I also understand the notation of why $\int_C f(x,y)ds=\int^{t=b}_{t=a}f(x(t),y(t))\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}dt$.

But what I'm stuck on, is just that my textbook just says"sometimes, you'll do a line integral with respect to $x$ only or with respect to $y$ only and that sometime you can combine them to see an expression like this: $\int_C P(x,y)dx+Q(x,y)dy$ ". So my question is about this part. What does doing a line integral with just $dx$ or just $dy$ mean, and what does it mean when they are combined into one integral. Is it saying something like "as you trace out the curve, you acquire this much mass in the x direction and this much mass in the y direction?"

EDIT: It took some time, but I found a similar post with very helpful answers! Here it is: Interpreting Line Integrals with respect to $x$ or $y$.

The second answer there is what I was looking for! But now it makes me think...the way my textbook presented it, it seemed like the textbook was trying to say:" $\int_C f(x,y) ds$ calculates the area of the curtain. And sometimes the $ds$ is broken into two parts $dx$ and $dy$ and calculated as $\int_C P(x,y)dx+Q(x,y)dy$." So the book made it seem like there's always some equality, as in $\int_C f(x,y) ds=\int_C P(x,y)dx+Q(x,y)dy$ for the right choice of $P$ and $Q$." And am I to understand that the right $P$ and $Q$ to make this work are the functions that are projected onto the $xy$ plane and the $yz$ plane? Is that saying "the area of the curtain = all the area you'd see looking at it from one direction + all the area you'd see looking from the other direction?" Is this correct?

Answer 1

I find it more intuitive to think of a function that returns a vector $F(x,y) = (P(x,y),Q(x,y))$ as its output.

The vector fuction represents a force field. That is, at every point $(x,y)$ there is some force acting on you. The force could be gravity, and the vectors point "downhill." The degree that you are walking uphill you will spend energy. When you move downhill you get some of that energy back.

$\int P(x,y) dx + Q(x,y) dy$ represents the energy spent or the change in potential energy

When you are have a parametrized path $r(t) = (r_x(t), r_y(t)),$ then $ (dx, dy)$ will become functions of the parameter $(r_x'(t), r_y'(t))\ dt$

Then you can say $\int P(x,y) dx + Q(x,y) dy = \int F(x,y)\cdot dr$