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### Course: Multivariable calculus>Unit 4

Lesson 1: Line integrals for scalar functions

# Introduction to the line integral

Introduction to the Line Integral. Created by Sal Khan.

## Want to join the conversation?

• Can anyone explain why this would be called a queue integral? My prof said something like "If line integrals were named in modern times it would've been called a queue integral."
• It's a joke regarding the colloquial usage of both words today. It used to be that people would form "lines" in shops and when asked to be orderly. Now, it's more common - nearly ubiquitous in the UK, not so much in the US - to hear "if you could please form a queue."
Mathematically, it's nonsense - the idea of a line in math is very different from the idea of a line/queue of people, and therein lies the humour.
• Can You please tell me what is the difference between "delta x" and "dx"? (mentioned between -)
• Delta x is the change in x, with no preference as to the size of that change. So you could pick any two x-values, say x_1=3 and x_2=50. Delta x is then the difference between the two, so 47.
dx however is the distance between two x-values when they get infinitely close to eachother, so if x_1 = 3 and x_2 = 3+h, then dx = h, if the limit of h is 0.
I guess basically dx is the special case of delta x when delta x is infinitely small.
• At , why is Sal using the Pythagorean theorem? Essentially he wants to measure the slope at that point right? That's what he means by 'arc length'. So why not differentiate the parametric equations right there which will give you the dS for that arc length? Because that's ultimately what he does in the end of the video at and but by then its mixed in with the Pythagorean theorem. It doesn't make sense to me because it doesn't seem consistent with how we're suppose to take the derivative of a parametric equation from the Taking Derivatives playlist.
• On one hand I can sympathize with your position – Sal can be excruciatingly detailed at times – especially when the listener already has a good understanding of what is going on, like you seem to have here. However, not all listeners, in fact most I dare say, are such that when they get to this Line Integral introduction video, they are seeing the concept for the first time and usually do not yet have a well-developed sense of mathematical intuition about it nor the concepts leading up to it. Apropos, Sal takes a short step back to remind them of what they did in the Integral Calculus track, namely Arc Length (https://www.khanacademy.org/math/integral-calculus/area-and-arc-length-ic/arc-length-ic/v/arc-length-formula) and use that to show that something similar is going on here. This is a common “building on what you already know” style of teaching. Here is another example of a similarly done introduction to Line Integrals, though with less depth of the details (you need to page down to get to the part I am talking about: http://tutorial.math.lamar.edu/Classes/CalcIII/LineIntegralsPtI.aspx. I have collected some 40 Calculus Textbooks in my time, and I would say about a quarter take this “back story” approach (these would be the texts which are aimed at those students who only need calculus to graduate – the texts written for those students who are actually studying math do not repeat themselves as much, since, mathematicians are in some sense, lazy – say something once, why say it again?

Now, on the other hand, have you ever gone back to watch one of Sal’s algebra videos? (Try it just for fun). Some concepts that should be obvious by that point are still explained in detail even though they are not directly about the video topic at hand, yet, their inclusion does not detract from the meaning of the message (they just make you roll your eyes and think “shouldn’t students know this by now?”. IMHO, it is the same thing with this video – we are reminded how dS is related to the Pythagorean Theorem, and how we have previously seen an example of it. This is done as a tool to boost the student’s intuition. From my assessment of the video, there is no confounding of this “back story” of the relation between dS and the Pythagorean Theorem with the introduction of the Line Integral formula.

So, perhaps, and hopefully, your situation is that you just found the “back story” a bit of a waste of your time (because you already knew what to do), and hopefully, it is not the case that the Pythagorean explanation truly "did not make sense” to you. If it really was confusing to you, that may mean your intuition in this area isn’t what you thought – but I doubt that, it sure seems from the details in your question that you already had a good idea of what was going on and it more likely that you are really just a bit annoyed at the “Pythagorean trip down memory lane.”

Anyway, something to think about, or not.
• In the final formula, why is x(t) and y(t) used instead of g(t) and h(t)
• It is common to use the variable name to refer to the component functions when you parametrize a curve, so x = x(t) and y = y(t). Here x(t) = g(t) and y(t) = h(t) defined earlier in the video.
• I was just wondering, why are we only taking the single integral and not the double integral for line integrals?
• Double integrals are for integrating over surfaces. Single integrals are for integrating over lines.
• can line integral be path independent....i think not!...but then hat are the function in thermodynamics that are path independent??i.e so called functions of state!!
• Line integrals from one point to a different point can be path independent, but this only holds for some functions. Imagine a map where you can read the height at any location. If you start at positions A and goes to position B, the change in height is independent on the path you choose to go from A to B. The change in height can be expressed as a line integral over the chosen path.

The term for the functions that have this property is "Conservative vector fields".
• What does this symbol mean ∮ ? Also at , how would it work to use parametric equations for the positions?
(1 vote)
• If one considers a line integral along a closed path, the symbol `∮` is sometimes used to emphasise that the path is closed.

At the time you reference, he does use a parametric equation for the position.
• If ∮ means line integral of a closed path then how do you evaluate line integral of a open path

Is it compulsory to use parametric equation
• An open path is just your typical line integral that you know and love, the only difference for the closed version is the endpoints a and b coincide.
If parameterization makes evaluating the integral easier, use it.
This wiki page has some nice graphics:
https://en.wikipedia.org/wiki/Line_integral#Vector_calculus
• `z = f(x,y)` forms a surface because the x-y plane is already a surface and we're just manipulating the values at each point to give us a z value. Parameterizing x and y just means that they're the inputs of the function f(x,y). An input is often called a parameter.