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

[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...

 

[SteamKing]

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.

 

[mrcleanhands]

oook so it is actually a curtain area still except this time it's the relevant area under the plane z=2x

 

[SteamKing]

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.

 

[HallsofIvy]

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.