What is the interpretation of a line integral with a 2D function?

In summary, a line integral for a function in 3D space above a curve represents a curtain-like space, similar to finding the area under a 2D curve. However, when dealing with a function like 2x, it is important to consider the height of a second curve above the original curves, as shown in the attached example. This height can be thought of as a fence, following the path of the original curves and with a height of 2x. It is essential to understand that while "area between two curves" and "area above a path" are possible applications of integrals, they are not the integrals themselves and may have different specific applications.
  • #1
mrcleanhands
When I think line integral - I understand when I'm taking a line integral for a function f(x,y) which is in 3D space above a curve that the integral is this curtain type space, just like if you had a 2D function and you find the area under the curve, except now it's turned on its side and it's 3D and two of the functions are changing, not one.

However, working my way through a calc book I've got to this example... Now I'm a little unsure of the interpretation. Since the function in the integral is 2x I assume it's a 2D function. Since an integral is like multiplying changing functions I thought it must be the area between 2x and x^2 from x=0 to x=1 but I see if I integrate that way I get nothing like what I get when I do it through line integration...
 

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  • #2
With a line integral such as the one in the attachment, think of the 2x as giving the height of a second curve above the curves C1 and C2. For example, the height of the second curve is 0 at (0,0) and 2 at (1,1), and it stays constant at 2 until reaching (1,2). The line integral represents the area between C1 and C2 and this second elevated curve. In other words, think z = 2x.
 
  • #3
oook so it is actually a curtain area still except this time it's the relevant area under the plane z=2x
 
  • #4
Or its the area of a fence. The fence follows the path described by C1 and C2, and the height of the fence is 2x.
 
  • #5
Note that "area between two curves" and "area above a path" are possible applications of integrals, not the integrals themselves. They can be useful ways of thinking about integrals but you should keep in mind that specific applications of integrals might be quite different.
 

1. What is a line integral with a 2D function?

A line integral with a 2D function is a mathematical concept that involves calculating the cumulative effect of a 2D function along a specific path or line in the 2D plane. It is used to measure the total value of a function over a given curve or path.

2. How is a line integral with a 2D function different from a regular integral?

A line integral with a 2D function is different from a regular integral in that it involves integrating a function over a specific path or curve, rather than over a specific interval. This means that the value of the integral can vary depending on the path chosen.

3. What is the physical interpretation of a line integral with a 2D function?

The physical interpretation of a line integral with a 2D function is that it represents the total work done by a force field along a specific path. It can also be visualized as the total area under the curve of the function, similar to a regular integral.

4. How is a line integral with a 2D function calculated?

A line integral with a 2D function is calculated by dividing the given path into small segments and approximating the function value at each point on the path. The values are then summed up and multiplied by the length of each segment to get an approximation of the total value of the integral. This process is repeated with smaller and smaller segments to get a more accurate result.

5. What are some real-world applications of line integrals with 2D functions?

Line integrals with 2D functions have many real-world applications, including calculating the work done by a force in physics, finding the mass of a thin wire or string in engineering, and calculating the total fluid flow in a pipe or channel in fluid mechanics. They are also used in computer graphics to create 3D images and in economics to analyze the total cost or revenue of a product or service.

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