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Worked example: Riemann sums in summation notation

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Here we express the approximation of the area under a curve in sigma notation. Created by Sal Khan.

Video transcript

Voiceover: What I want to do in this video, is get a little bit of practice trying to approximate the area under curves, and also get a little bit more familiarity with the sigma notation in this context. So what we have here, we have the graph of f of x is equal to one plus 0.1 x squared, that's this curve right over here, and then we have these rectangles that are trying to approximate the area under the curve, the area under the function f, between x equals zero, and x equals 8, and the way that this diagram, or the way that we are attempting to do it, is by splitting it into four rectangles, and so we can call this rectangle one, this is rectangle two, rectangle three, and rectangle four. And each of their heights, let's see the interval, it looks like they each have a width of two, so they are equally spaced, so we go from zero to eight, and we split into four sections, so each has a width of two, so they're each going to be two wide, so that's two, that's two, that's two, that's two. And their height seems to be based on the midpoint, so between the start, between the left side and the right side of the rectangle, you take the value of the function at the middle value, right over here, so for example, this height right over here looks like f of 1, this height right over here looks like f of 3, this height of this rectangle is f of 5, this height right over here is f of 7. So given the way that this has been constructed, and we want to take the sum of the areas of these rectangles as an approximation as the area under this curve, how would we write that as sigma notation? And I'll get us started, and then I encourage you to pause the video and try to finish it. So the sum of these rectangles, we could say it's the sum of, so we'll have n equals one to four, 'cause we have four rectangles. And I encourage you to finish this up. Actually, just write it in terms of the function, use function notation, you don't have to write it out as one plus .01 times something squared. So I'm assuming you've had a go at it. So for each of these, so for the first rectangle over here, We're gonna multiply two times the height, so the height right over here is one, and this is the first rectangle, so you might be tempted to say times f of n, but then that breaks down as we go into the second rectangle. The second rectangle, the two still applies. This two is the width of the rectangle, but now we want to multiply it times f of three, not f of two, so this f of n isn't going to pass muster. And so let's see how we want to think about it. So when n is one, two, three, four, we're gonna take f of... So f of n, I shouldn't say f of n, we're gonna take f of something, so here this first one, we're gonna take f of one, then here, for the second rectangle, we're taking f of three for the height, for the third rectangle we're taking f of five, and then for the fourth rectangle, we're taking f of seven. So what's the relationship over here? Let's see, it looks like if you multiply by two and subtract one, so two times one minus one is one, two times two minus one is three, two times three minus one is five, two times four minus one is seven. So this is two n minus one, so the area of each of these rectangles, the base is two, and the height is f of two and minus one. So that hopefully makes it a little bit clear, kind of mapping between the sigma notation and what we're actually trying to do. And now let's just for fun, let's actually try to evaluate this thing. What is this thing going to evaluate to? Well, this is going to evaluate to two times f of, when n is equal to one this is one, f of 1, plus two times, when n is two, this is gonna be f of, two times two minus one is three, f of 3, when n is three, this is gonna be two times f of 5, when n is four, this is going to be two times f of seven, four times two minus one is seven, f of seven. And so that is going to be, well we're gonna have to evaluate a bunch of these things over here, so let me actually, let me clear this out so I have a little bit more real estate. I'm feeling this might get a little bit messy now. So this is going to be, actually we could factor out a two. So this is going to be equal to two times f of one is one plus 0.1 times one squared. So it's one plus 0.1, let me color-code it a little bit so we can keep track of things. So this right over here is 1.1, so one plus 0.1 is 1.1, this right over here, f of 3, so that's one plus .1 times three squared, nine, so one plus .9, so it's 1.9. And then, let's see, this one right over here, f of 5, is gonna be one plus, see five squared is 25, times .1 is 2.5, so one plus 2.5 is gonna be 3.5. And then finally, f of seven, is going to be one plus .1 times seven squared, so this is 49 times .1 is 4.9 plus one, so plus 5.9. And so what is this going to be equal to? So let's see, 1.1 plus 1.9, these two are going to sum up to be equal to three, and then these two are going to sum up to be, let's see, if we add the five, we get to 8.5, and then we add the .9, we get to 9.4, so plus 9.4. Did I do that right - three plus five is eight, .5 plus .9 is 1.4, yep, and so this is going to be, so once again, we have the two times it all, so this is going to be equal to two times 12.4, which is equal to 24.8, which is our approximation. Once again, this is just an approximation using these rectangles of the area under the curve between x equals zero, and x equals eight.