Question

Hello, I am a bit confused as how line integrals can exist in 2D.

For 2D, we are dealing with two variables: x and y.

For 3D, we are dealing with three variables: x,y and z=f(x,y), such that f(x,y) = x+y. Since line integrals use the x and y components as inputs and f(x,y) = z as the output, should not line integrals exist in 3D always?

Answer 1

Recall what you mean by and integral over a real line: It tells you the area the function encases between it and the real line.

Now what would line integral in 2D mean? It just means that now you have a random line (can be seen as a copy of the real line sitting inside 2D, i.e. translation of the 1D vector subspace of R; if you're aware of vector spaces) and now you integrate the function over this line (despite R2 having two variables any line in R2 has only one free parameter, example any line has the equation ax+by=c and hence x depends on y and thus the line can be parametrized by one parameter only).

Similarly you have line integrals in 3D also, with the equation of line being   which is again one parameter family. Now you integrate to get the area beneath the curve and this line.

So when you plug in the value f(x,y) and integrate it over a line, the z=f(x,y) just tells the height of the curve of f at the point (x,y).