Interpreting Line Integrals: Is My Understanding Correct?

In summary, a line integral involves picking points off of a curve, putting them through a function, and summing the individual values to infinity. This can be re-written as a normal integral with a transformation applied to the x-axis. This means finding the area between the function and the transformed x-axis. The conversation confirms this interpretation and provides a visualization for a simple example.
  • #1
chaoseverlasting
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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

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

[tex]I'=\int f(g(x))[/tex] (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?
 
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  • #2
That's pretty much it. A visualization for a very simple example of f:R2->R2 and g:[0,1]->R2 is given here.
 
  • #3
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!
 

What is a line integral?

A line integral is a mathematical concept used to calculate the total value of a function along a curve or line. It involves breaking the curve into smaller segments and adding up the values of the function at each point along the curve.

How do I interpret a line integral?

To interpret a line integral, you need to understand the function being integrated, the curve or line being integrated over, and the direction of the integration. The result of the line integral represents the total value of the function along the curve in the given direction.

What is the difference between a line integral and a regular integral?

A line integral is a type of integral that is performed over a curve or line, while a regular integral is performed over an area or volume. Line integrals also take into account the direction of integration, whereas regular integrals do not.

What is the significance of the direction of integration in a line integral?

The direction of integration in a line integral determines the positive and negative contributions of the function along the curve. It can also affect the final value of the integral, as reversing the direction of integration can result in a negative value.

Can line integrals be used in real-world applications?

Yes, line integrals are commonly used in physics and engineering to calculate quantities such as work, electric or magnetic fields, and fluid flow. They are also used in computer graphics to create smooth curved lines and surfaces.

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