Interpreting Line Integrals: Is My Understanding Correct?

[chaoseverlasting]

I have a rough idea of what a line integral is, please correct me if I am wrong:

If a function y=f(x) is being integrated over a curve c=g(x), what we are doing is picking points off of the curve c, putting them through f(x) and summing the individual values that we get to infinity (or sum of infinitely small values between a given set of limits). In other words, the integral

$$I=\int f(x)$$ (I is a Line Integral) from a to b over a curve c may be re written as:

$$I'=\int f(g(x))$$ (I' is a normal integral) from a to b, where g(x) is the curve c.

What this really means is that we are first applying a transformation c=g(x) to the x axis, and then defining a curve y=f(x) on this transformed x-axis and then finding the area between f(x) and the transformed x-axis between the limits a and b. Am I right in this interpretation?

 

[slider142]

That's pretty much it. A visualization for a very simple example of f:R²->R² and g:[0,1]->R² is given here.

 

[chaoseverlasting]

Thank you for confirming it and for the link as well. I am so happy I am actually beginning to understand the language used by mathematicians! Sounds like Greek sometimes!