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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....

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....