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Studying for a test? Prepare with these 5 lessons on Riemann sums.
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Video transcript
- [Voiceover] Right over here we have the graph of F and then we have four different expressions. What I encourage you to do is pause this video and see if you can figure out which of these expressions would give the largest quantity, the second largest quantity, the second smallest quantity and the smallest quantity. I'm assuming you have paused the video and you have given an attempt. Now let's work through this together. This first expression right over here, we're taking the sum from I equals zero to nine. We're actually taking the sum of 10 things, because we're taking the zero thing, first, second, third, all the way up to nine. So this is actually going to be the sum of 10 things because we're starting at zero. We start at F of negative five plus zero. We're saying negative five, F of negative five plus zero. That's this height right over here. That's that height right over there. We're going to take that, and then when I equals one it's going to be negative five plus one, which is negative four. It's that height right over there. Then negative five plus two when I equals two. It's going to be that height right over there. We're essentially going to sum up all the way. This is going to be negative five, negative five all the way up to, negative five plus nine is going to get us all the way to four. It's going to be all the way over there. All the way over there. You might be guessing, "Well, how do I relate this? "They've already kind of made us think "that we're going to somehow relate this "to area somehow, but how do we "actually make that relationship?" Because right now, as this is written, it's just giving us essentially a bunch of the values of the functions at different points. I guess you could say it's a lot of these heights, right over here. But one thing that might jump out at you is you could construct rectangles, all that have width one, and so if you multiply the height times the width, the area is going to be the same thing as the height. If we put a one times one right over here, this makes it very clear that you're taking the height times the width of this rectangle and then this rectangle. You essentially have a bunch of left-handed rectangles that you could imagine are trying to estimate this bluish area that was shaded in. It's clearly going to be an underestimate, because it's giving up these areas. It's giving up those areas, right over there. All of these rectangles are sitting, they're either just touching or they are below the actual function. Let me just write this right over here. This is going to be an underestimate of the area of this blue area. Now let's think about what this one is, here. This is F, it's the same thing that we're taking the sum of. We're starting at I equals one and we're going to 10. Once again, 10 things. Negative five plus one is negative four. F of that is this line, right over here. Is that line right over there. It looks like we're taking right-handed rectangles, because you could say times one, times one would be the area. Obviously, if we multiply by one, we're not changing the value of this expression. That would be the area of this first right-handed rectangle. Then when I is equal to two, it's going to be F of negative three, and so you're looking at this one right over here. I can, let me draw at least this part of it. It's going to look something like this. It's going to look something like this, where now we're dealing with right-handed rectangles, and these are clearly an overestimate. These are right-handed rectangles. You're going to go all the way, when I is equal to 10, negative five plus 10 is five F of five. That's this line right over here, or this length right over here, F of five. Of course, we're multiplying it by one. It's going to look like that, and we could keep going. I think you get the general idea now. These are all going to be right-handed or these are all right handed rectangles that I've drawn. These are going to be an overestimate of the area, because they all have this little extra, they all have this extra region, right over here. These are going to be an overestimate. Now let's think about this one, right over here. This is, we're going to start at I equals one and we're going to go to 20. It looks like we're going to do rectangles instead of width one, we're going to width one half. Once again, since we're starting at I equals one, these look like right-handed rectangles again. This is, that we're going to use to estimate. Let me do this in a color that you can actually see. I'll do this in orange. I'll do this in orange right over here. The first one is going to be negative five plus one half. It's going to be this, F of that, which is going to take you right over there. Then you're going to multiply it times the width, which is one half. Now we have twice as many, twice as many right-handed rectangles. It's going to look like this. Twice as, and I won't do all of them because it takes some time. We're going to have twice as many right-handed rectangles. It's still going to be an overestimate, but it's going to be less of an overestimate than this one over here. Because this one over here, you had all of this extra green space above the function. Now we have a lot less extra space, a lot less extra space above the function. It's a better estimate, but it's still going to be an overestimate, because at least for this function right over here, because at least over this interval where the function is increasing, the right-handed rectangle is giving us an overestimate. But this is a little bit more precise because we're using narrower rectangles to estimate. Overestimate but less so, but less so than this one right over here. This right over here, this is a definite interval, from negative five to five of F of X, DX. You can imagine, this is essentially the limit as we take these widths to be smaller and smaller and smaller. We essentially end up having essentially an infinite number. We approach an infinite number of these right over here. This is the actual area. This is what's actually depicted in blue. This is the actual area. If I were to list this from largest to smallest. The biggest overestimate, this is the biggest overestimate, right over there. This is still going to be an overestimate but it's a little bit more precise because we have more rectangles. I'll put that two. This is the actual area. I'll put that three. Then this is actually an underestimate. I would rank this four. This is the largest of the values, and this is going to be the smallest of the values.