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Integrate byline understanding

in this video we're going to talk about line intervals or sometimes called path intervals this video was part of my playlist on vector calculus and i'll link that playlist and all of my other course playlists down in the description now in this video we're going to do two major things first we're going to introduce the idea of what is a line integral what's the big idea what problem are we trying to solve and then secondly we're going to work towards coming up with a formula that lets us compute line intervals and then in later videos we see examples and other applications of line intervals again links to those in the description now before we jump into the math before we see any graphics here's one way to visualize the idea of a line interval imagine it snowed and there's snow banks everywhere outside some spots the snow is higher and at some spots the snow is less high and you want to go and walk out into the snow shoveling as you go the question is how much do you have to actually shovel when you go out and walk from your back door well it depends on the path you take so it certainly depends on how far you go for example if you go further there's going to be more snow that you're having to shovel as you go away but it also depends on where you go with the length that you decide to walk for example if you walk a lock through big snow drifts that are really deep then you'd have a shovel a lot more snow so what the line integral captures is this notion of an accumulation in this case of snow above a particular curve okay let's see a little bit more concretely what i mean with some mathematics formalism let's just begin in the simpler scenario we know well a simple function f of x from one variable to another and the important thing is if i look down at the domain well the domain i can think of is just some line segment on the x-axis this is kind of like a curve that's just constrained to be only the x-axis and then the f of x tells me the height above that line segment and then if i'm interested in figuring out what is the area well we know the answer from single variable calculus integration was the answer to the question what is the area above some interval and underneath some function f of x okay so now let's go up a dimension now my input is two-dimensional it's the x-y plane and in the x-y pane i have not a line segment but some other curve in this case it is a circle i'm going to have a height above that a third dimension in a moment but right now i'm saying my inputs is going to be this x y plane...