Calculus, all content (2017 edition)
Generalizing the technique of approximating area under a curve with rectangles. Created by Sal Khan.
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- At7:25, he says "X sub i-1." What does "sub" mean?(35 votes)
- Just wondering. How is this different from a right Riemann sum or a midpoint Riemann sum?(19 votes)
- The right hand sum is where instead of making f(x) the value from the left side of the rectangle, it's the right side. Midpoint is where you take f(x) where x is in between the left and right endpoints of dx.(1 vote)
- where did that "i" come from and what is it's purpose?
i get that the bottom of the Sigma is the starting rectangle and the top is the finish ("N") but then the i re-appears in subtext of x. any help?(14 votes)
- "i" is just a letter for number of rectangles, starting with 1 and ending with n. As for the subtext, it is just pointing out that for rectangle 1 (i=1), x(i-1) (in this case x0) is used and for rectangle 2 (i=2), x(i-1) (in this case x1) is used... and so on until rectangle n (i=n), where x(n-1) is used. Hope that helps.(24 votes)
- I am vaguely familiar with Riemann's work and, from what I understand, wasn't the stuff he was doing much, much more advanced than this? Why would he waste his time doing these sums?(8 votes)
- Although integrals had been studied long before Riemann, it was his work that led to a far more rigorous approach and formal definitions. So, he was advancing integral calculus and making it far more rigorous. So, this was not unimportant work even if it seems too simplistic nowadays.
Though, of course, Riemann was one of the greatest mathematicians who ever lived and accomplished much more sophisticated feats than formalizing integrals.
It should be noted, however, that not all integrals are compatible with Riemann's work with sums. But that is a more advanced topic.(32 votes)
- What will happen if we take the limit as delta x approaches 0 ? Won't it give us the exact area?(10 votes)
- Yes! - and this video is meant to give you the background intuition for the subsequent videos where that limit will be taken giving a more formal definition of the definite integral.(19 votes)
- What is that big "E", why does he call it a sum, what is all of the notation around it, and if Sal has done that in another video, which one is it? What's it called?(4 votes)
- That "big E" is actually not an "E." It is the Greek letter "Sigma," which is the equivalent of the English letter "S." It stands for "Sum." If you can begin to see it as a "Sigma," or especially as the letter "S," the notation for integration will begin to make a little more sense.
If you are going to study more advanced mathematics (or especially physics, which uses this math), it would be advantageous for you to become familiar with the Greek alphabet, both the upper and lover case letters.
Hope this helps.(28 votes)
- At8:05, he says that x sub i-1 is for the left side, how should it be written if you want to use the midpoint or the right side?(9 votes)
- For right endpoints, the notation is simply x sub i,
left endpoints are, as Sal points out, x sub i-1,
midpoints are x sub i* (star), which looks like x sub i with a star directly above i.(8 votes)
- Why don't we use simple integration method to find the area?(2 votes)
- This section on Riemann sums covers the theoretical background that led to the development of integration. In this section you should gain an appreciation of what integration is by understanding the concepts of how it came to be.(21 votes)
- Why not just make it the sum of f(x) when i=0 to n? Since the first term is x sub 0. obviously the way he did it works with i-1 but it just seems more complicated.(6 votes)
- if you started at i=0 then you would have to do away with the "i-1", If we did things that way, we would need to change the upper limit of the sum to n-1 to ensure that we didn't take the height of the rectangle n+1 (which would have a height of f(n)). We could choose to change the upper limit but doesn't really capture the idea of a complete domain form a to b as well as 1 to n.(7 votes)
- What's the formulas for the right and middle Riemann sums?(8 votes)
- What I did to figure that out was draw box 1. The height is now x sub 1. the width is delta x, of course because that didn't change. So now, like what Sal did, I wrote the math notation for the sum of the boxes, the sum (sigma) (from i=1 to n) of the height (f(x sub i)) times the width (delta x).(3 votes)
In the last video, we attempted to approximate the area under a curve by constructing four rectangles of equal width and using the left boundary of each rectangle, the function evaluated at the left boundary, to determine the height, and we came up with an approximation. What I want to do in this video is generalize things a bit using the exact same method, but doing it for an arbitrary function with arbitrary boundaries and an arbitrary number of rectangles. So let's do it. So I'm going to draw the diagram as large as I can to make things as clear as possible. So that's my y-axis. And this right over here is my x-axis. Let me draw an arbitrary function. So let's say my function looks something like that. So that is y is equal to f of x. And let me define my boundaries. So let's say this right over here is x equals a. And this right over here is x equals b. So this is b. And I'm going to use n rectangles, and I'm going to use the function evaluated at the left boundary of the rectangle to determine its height. So, for example, this will be rectangle one. I'm going to evaluate what f of a is. So this right over here is f of a. And then I'm going to use that as the height of my first rectangle. So just like that. So rectangle number one looks like this. And I'll even number it. Rectangle one looks just like that. And just to have a convention here, because I'm going to want to label each of the x values at the left boundary, so we'll say a is equal to x0. a is equal to x0. So we could also call this point right over here x0, that x value. And then we go to the next rectangle. And we could call this one right over here, this x value, we'll call it x1. It's the left boundary of the next rectangle. If we evaluate f of x1, we get this value right over here. This right over here is f of x1, so it tells us our height. And we want an equal width to the previous one. We'll think about what the width is going to be in a second. So this right over here is our second rectangle that we're going to use to approximate the area under the curve. That's rectangle number two. Let's do rectangle number three. Well, rectangle number three, the left boundary, we're just going to call that x sub 2. And its height is going to be f of x sub 2. And its width is going to be the same width as the other ones. I'm just eyeballing it right over here. So this is rectangle number three. And we're going to continue this process all the way until we get to rectangle number n. So this is the n-th rectangle right over here, the n-th rectangle. And what am I going to label this point right over here? Well, we already see a pattern. The left boundary of the first rectangle is x sub 0. The left boundary of the second rectangle is x sub 1. The left boundary of the third rectangle is x sub 2. So the left boundary of the n-th rectangle is going to be x sub n minus 1. Whatever the rectangle number is, the left boundary is x sub that number minus 1. And this is just based on the convention that we've defined. Now, the next thing that we need to do in order to actually calculate this area is think about what is the width? So let's call the width of any of these rectangles-- and for these purposes, or the purpose of this example, I'm going to assume that it's constant, although you can do these sums where you actually vary the width of the rectangle. But then it gets a little bit fancier. So I want it equal width. So I want delta x to be equal width. And to think about what that has to be, we just have to think, what's the total width that we're covering? Well, the total distance here is going to be b minus a, and we're just going to divide by the number of rectangles that we want, the number of sections that we want. So we want to divide by n. So if we assume this is true, and then we assume that a is equal to x0, and then x1 is equal to x0 plus delta x, x2 is equal to x1 plus delta x, and we go all the way to xn is equal to xn minus 1 plus delta x, then we've essentially set up this diagram right over here. b is actually going to be equal xn. So this is xn. It's equal to xn minus 1 plus delta x. So now I think we've set up all of the notation and all the conventions in order to actually calculate the area, or our approximation of the area. So our approximation, approximate area, is going to be equal to what? Well, it's going to be the area of the first rectangle-- so let me write this down. So it's going to be rectangle one-- so the area of rectangle one-- so rectangle one plus the area of rectangle two plus the area of rectangle three-- I think you get the point here-- plus all the way to the area of rectangle n. And so what are these going to be? Rectangle one is going to be its height, which is f of x0 or f of a. Either way. x0 and a are the same thing. So it's f of a times our delta x, times our width, our height times our width. So times delta-- actually I can write as f of x0, I wanted to write-- f of x0 times delta x. What is our height of rectangle two? It's f of x1 times delta x. What's our area of rectangle three? It's f of x2 times delta x. And then we go all the way to our area. We're taking all the sums, all the way to rectangle n. What's its area? It's f of x sub n minus 1. Actually, that's a different shade of orange. I'll use that same shade. It is f of x sub n minus 1 times delta x. And we're done. We've written it in a very general way. But to really make us comfortable with the various forms of notation, especially the types of notation you might see when people are talking about approximating the areas or sums in general, I'm going to use the traditional sigma notation. So another way we could write this, as the sum, this is equal to the sum from-- and remember, this is just based on the conventions that I set up. I'll let i count which rectangle we're in, from i equals 1 to n. And then we're going to look at each rectangle. So the first rectangle, that's rectangle one. So it's going to be f of-- well, if we're in the i-th rectangle, then the left boundary is going to be x sub i minus 1 times delta x. And so here, right over here, is a general way of thinking about approximating the area under a curve using rectangles, where the height of the rectangles are defined by the left boundary. And this tells us it's the left boundary. And we see for each, if this is the i-th rectangle right over here, if this is rectangle i, then this right over here is x sub i minus 1, and this height right over here is f of x sub i minus 1. So that's all we did right over there times delta x. And then you sum of these from the first rectangle all the way to the end. So hopefully that makes you a little bit more comfortable with this notation. We're not doing anything different than we did in this first video, which was hopefully fairly straightforward for you. We have just generalized it using a little bit more mathy notation.