What is a LINE INTEGRAL? // Big Idea, Derivation & Formula

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

in particular this circle in the x y plane i can talk about this a bit more generally by parameterizing a generic curve which we call c the way i parametrize curves is i write it r of t is going to be an x component g of t in the i hat direction and then a y component h of t in the j hat direction and if i am parameterizing it like this i need to specify some domain of my parameter t so i say t is in some interval a up to b in this specific

example my functions might be 2 cosine of t and the x and 2 sine of t in the y and my interval might be 0 to 2 pi but you can do this for any curve the idea of parameterizing something means that you've got this single variable input t but that what you get out of it is this two-dimensional thing this r of t and indeed note that the r of t i've written here is two dimensional i don't yet have a third component to it okay so now let me plot a new curve well

this curve has the same x and y components but instead of having a z component being zero it now has a z component being some arbitrary height and i can call this height f of x and y it's some function of x and y now if i focus specifically on the x and y that are along this curve i could substitute in that this is f of a g of t and of an h of t that is my third component is now just some other function of t and by the way a very natural

way to represent parametric curves is as i did earlier with my animation as time went on more and more of the curve was drawn this is a great way to represent a parametric curve now it might be the case that the function f of x y is actually defined on many more points than just the curve c for example when i was figuring out this animation i actually have my f x y being some paraboloid it is defined on many more points than just a curve the only parts of the function i care about are

those above this curve and then my final step is just to shade everything in so at any point along my curve c i just draw up until i hit the blue to the height f x y and that's what i get so the question is what is the surface area of this resulting shape what is that area now at this point in our calculus development we've seen a lot of times where we've tried to tackle a geometric problem like finding an area for example and we've used integration and the big idea has always been to

break it into small little pieces that's the big idea of integration take some complex problem you can't solve like what is the surface area of this thing break it into a much smaller thing you can solve and go from there so for example instead of this nice clear continuous what if i break it up into a bunch of smaller rectangular chunks this is an approximation for the original surface area i want to compute okay let me let me break out exactly how i got those if i start with my pair of curves my first step

is to take the time interval remember the t was your parameter here and it ranged between a and b take that time interval and break it into n different components and what you can see here is that in the graph i just plotted a lot of points and those just represent as i'm trying and drawing those curves what happens when my t is one nth of the way two ends of the way three ends of the way and so on okay and then the next step is i'm going to draw line segments between all these

points and notice i've done it twice first down in the circle i've taken my circle and approximated by a bunch of little line segments and then likewise on the blue curve as well and then the next thing i'm going to do is create a whole bunch of rectangles this is just one of those sample rectangles and these are just going to connect a bunch of these points but the big idea of integration has always been yes you can do this but then you just increase your end more and more so for example if i wanted

to have more points i've doubled the number of points or double the number of points again my rectangle gets smaller and it was never quite right because there's always like a little bit of a gap at the top like it never fit perfectly but when you get more and more and those gaps get smaller and smaller that's the big idea okay so let's return to the slightly larger one in fact i even want to zoom in on it now this rectangle i want to figure out what's its area because i'm going to add up the

area of all of these rectangles so i want to know the area of one of them well the height is just the function value at whatever particular point you're at indeed i'm imagining that i'm at the k point remember i have n in total so if i'm at some k in the middle there then i'm going to figure out what is the function at that k value so there's an x k and a y k there's kind of a bit of a choice here uh the notice the rectangle actually goes from the k point to

the k plus one point so i could have chosen the height of the k plus one point as well or i could have chosen halfway between those there's some options i've sort of chosen the left endpoint approximation in the limit as n goes to infinity it's not going to matter okay so that's the height and then what about the base the base now i call delta s k s is the symbol for arc length so basically i'm saying well i've got a little arc length down here on the bottom i'm going to call that delta

sk we're going to investigate it more in a moment but right now i'm just going to call it sk and then well what's my rectangle it's these two things but just multiplied together it's got the height f x k y k and then it's got the width the delta sk and together that is my delta a k my little area my k area okay so now we're ready to finally define what we mean by a line integral so here it is it's a fancy set of symbols and notice this is a new symbol that we

have not seen before this is not an integral from a up to b it's an integral with a subscript of c and that just means a line integral over a curve that's how i interpret this and then for the integrand i put my f of x y analogous to my normal heights that i put in my integrand and then and then i wrote d s for my little infinitesimal increase in the arc length s okay so this is a new symbol how is it defined well it says let's look at that approximation and take the

limit when i sum up all of those different things that is the standard riemann integral definition but for this situation it's a limit as n goes to infinity you break it up into as many pieces as possible and then you sum up all of those delta aks and that is my formal definition okay so this is pretty good as in i've come up with a definition that's analogous to any number of definitions earlier in calculus for computing areas and volumes of things but as you'll recall limits of sums are not easy to compute so this

might be the formal definition but i also want to come up with a formula that lets me compute this limit of the sum easier without actually having to do a limit of a sum because that is very very tedious i want a quick computation so let's actually return just back a little bit in our definition where we had the area of a rectangle we decided that its height was f x k and y k and that its base was the delta s k i want to do better here so let me focus on specifically the

delta s k well if i imagine what's happening in the base this change in arc length there's a change in the x and there's a change in the y and so the change in the arc length by pythagoras is just going to be the same thing as well the square root of the sum of the squares the the change in the x squared that is plus the change in the y squared that is that square root it and so now i've managed to actually improve what my delta s k is okay so so now let's

go back up same story we had before delta ak exactly we just saw i've got my height my f but now my delta sk has been replaced by this pythagorean thing now i'm going to do a little bit of a trick on this delta sk business in fact it's the exact same trick that we did back in multi-variable calculus i'll make sure to link the video when we're coming up with a formula for the arc length of a curve back in multi-variable calculus the trick was i'm going to divide and multiply by the delta t

and so there's a delta t on the outside and then because of the square root there's delta t squared's on the bottom i just multiplied by one and then when i take the limit as n goes to infinity here a couple different things happen first the f of x k y k just turns into the the f of the x and the y at that specific point t so g of t and h of t so so the f changes from at a specific point to as a function of t and then more importantly if

you look at the delta x k over delta t as n goes to infinity this turns into the derivative of the x component the x component was called g so it turns into g prime and then likewise delta y k over the delta t that in the limit as n goes to infinity also turns into h prime so either way i get this new formula in terms of the derivatives this g prime squared and this h prime squared okay so if i just focus on this now that is what i'm going to integrate and so

i have a new formula for my line integral the line integral the integral with respect to some curve c of f x y d s is now going to be just the integral from a to b hey this is something i can do now this is one of my old single variable integrals this is an integral in terms of t so i can compute that anyways i plug it in f of g of t and h of t and the square root of g prime squared plus h prime squared so here's the point if you

know what your g of t is your h of t is and your f of t is and you know the bounds of your parametrization the a and the b well this is just an integral you plug everything in and you can compute it the way i think about this is all of that business with the square root the square root of g prime squared plus h prime squared dt all of that is just the expanded form of just the little arc length the little ds and so when i look at my picture i think

that i am trying to add up a whole bunch of little components my components have the height the f and then you have the width the little dx and that's what i think of this formula so in fact i don't go through the whole process of deriving this formula every time when i just look at this formula i think okay i'm trying to figure out the surface area it's just heights times little ds's and we previously analyzed little ds can be written with this big square root stuff okay what are we going to do over

the course of the next couple of videos well first i definitely owe you a concrete example of this so in the next video we're going to have actual functions listed and run through a process of computing a line interval in a specific example but then secondly i actually want to show you that there's many many more applications of line integrals than the specific one we saw in this video of computing this surface area indeed it turns out that the original curve c was two-dimensional in our example but you could have those curves being three-dimensional or

even higher and indeed the thing you might be caring about might not be a geometric property like a surface area there can be all sorts of other properties that you might be interested in that a line interval allows you to compute so all of that is coming up in future videos again playlist is linked down in the description so if you like this video give it a like if you have a question leave it down in the comments below and we'll do some more math in the next video