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Calculus: Areas, Riemann Sums, Definite Integrals


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About this Lesson

  • Type: Video Tutorial
  • Length: 13:40
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 148 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

Calculus (279 lessons, $198.00)
Calculus: Basics of Integration (14 lessons, $23.76)
Calculus: The Fundamental Theorem of Calculus (5 lessons, $9.90)

Taught by Professor Edward Burger, this lesson comes from a comprehensive Calculus course. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers limits, derivatives, implicit differentiation, integration or antidifferentiation, L'Hopital's Rule, functions and their inverses, improper integrals, integral calculus, differential calculus, sequences, series, differential equations, parametric equations, polar coordinates, vector calculus and a variety of other AP Calculus, College Calculus and Calculus II topics.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, "Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas" and of the textbook "The Heart of Mathematics: An Invitation to Effective Thinking". He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The "Journal of Number Theory" and "American Mathematical Monthly". His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

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The Basics of Integration
The Fundamental Theorem of Calculus
Areas, Riemann Sums, and Definite Integrals Page [1 of 1]
So, our mission is to figure out how to find the areas under a curve but above the x-axis. So, how to find this pretty yellow area for any kind of function f(x). That's our mission, and you saw last time that one way to attack this problem, from at least from a philosophical thinking point of view, would be to cut the world up into a whole bunch of rectangles, and then find the area of each of those rectangles, which is pretty easy to do, it's just base times height, and then sum all that up and you summed them up and then in fact, you would have an estimate for the very end of the curve. And, if you actually take thinner and thinner rectangles and put more and more rectangles in, you get closer and closer to the actual answer, and you sort of take the limit as you put in infinitely many rectangles, so I they get sort of instantly thin, than in fact, you would get the exact area. So, how do you actually execute this thing in practice? Well, let me try to write some stuff down here and at least inspire the notation that we are going to use.
So, what we want to do here, is capture the spirit of that previous picture and just draw it up here again for us, so we can always have this. We can write all over it, I hate to write on, we have such a pretty picture, I don't want to write on a pretty picture. I know I can, I mean, because it's my thing, I can do whatever I want, but you know, it seems so wasteful. So here's the function, this is function y = f(x). I'm glad this course is coming to an end by the way I can look at my pens are running dry. So, it's a good time to bring things to a close. And let's call this point here, let me call this lower case a, and I'll call this point here, lower case b, and the question is to find the area under the curve, but above the floor. So, between the curve and the x axis. And our idea, our strategy for this was to fill this thing up with a whole bunch of rectangles, to fill this thing up with a lot of rectangles, and I'll just draw some of them in here so you can sort of get a sense of this. Just stack rectangles in there, and then find the area of each of those rectangles, base times height, and the more rectangles you put in, the better the approximation. There you can see a lot of rectangles there, you see, I'll draw, I'll just color one generic one in, just so you can see, actually you can see that rectangle there's that rectangle. So, there's a generic one, but of course, there are a lot of them here.
Okay, now how do we approximate the area? Let's call the area, a capital A. So let's call the area, a capital A, A is the area. And what will that area equal? But we don't know what it equals, of course, if we knew what it equals, we'd be done, and we would be wasting your time right now. We're trying to figure that out, but it's approximately equal to the sum of the areas of the rectangles. I want to write that down, so the sum of the areas of the rectangles. And that makes sense, that's just sort of the observation that these rectangles altogether sort of fill up almost perfectly, this curvy thing, but not quite. So, it's approximately equal to that. Well, let's see, what are the sum of the areas of the rectangles? Well, that actually itself equals, so this equals what? Well, I'm going to sum all these things up, well, I'll say sum. So, sum the areas of the rectangles. Well, what are the areas of the rectangle? Well, base times height. So, I want to write that as height multiplied by base. Okay? All right, let's see if I can patch that up really fast for you. Okay, great. So, that produces the sum of the areas of the rectangles, I sum up base times height, for each of the rectangles, one, two, three, four, five, six... Base time height, base time height, base time height, base time height, base time height, base time height, add them all up... and I get an approximation to the actual retail value, the actual area. Okay, are you with me?
All right, now let's figure out what the base and the height are. Well, this equals, so sum, I'm not going to write the "um" anymore, so, let's right sum and what's the height? Well, I'm at a particular point sort of x, the height of the rectangle would be just the function, remember the function gives me height, this is going to be f(x). I have to multiply that by the base. Now what is the base? The base is a very tiny change in x. So, you can think of that as a delta x. So, if you really look at this carefully, I hope you see that this is exactly what I advertised, I see that the area is approximately equal to, I just sum up all the areas that are a lot of rectangles, where they have very, very tiny bases, small changes in x, and their height is just the value of the function there. This is just the area, and I sum it all up. Well, now if I let that small change in x get smaller and smaller and smaller and smaller. I get closer and closer and closer to the actual area. I sort of take the limit and I can think of that as the delta x becoming really, really, really small, and so in the limit, I might want to think of this as a dx. So, the derivative idea, where we went to the delta x to the dx, when we took the instantaneous for a change. And so, in fact, what we could write, this is just notation now, what we could write the following, I could write that the area would be equal to what I get when I sum up infinitely many of these rectangles. So I'm going to sum up the areas of the rectangles, f(x), and I'm going to write dx thinking about the fact that this is approaching zero, another very tiny change, instant change. Now, I have to take into account, that I'm starting at a and going to b, so I'll write that by putting a little a down here and then a little b up here. And, I'll write this for the area. Now, if you look at that, that should look extremely familiar because without those little a's and b's, well, that's just looks like, at least physically, the anti-derivative. Is it the anti-derivative? Well, it turns out the answer is, it really is the anti-derivative. That is to say, if you want to find the area under a curve of a particular function, what you need to do is look at the anti-derivative of that function. It's an absolutely amazing and surprising fact that somehow, by finding an anti-derivative, that gives rise to the exact area under a curve. Why in the world should we even believe this is true?
Let me try to inspire, at least in principle, why you can see roughly speaking why, by taking an anti-derivative, we get an area under a curve, because it's not at all obvious. So, I hope that this argument that I'm about to give will make some sense for you. But if nothing else, at least you can see that one can think about a reason why this does make some sense. So, let me draw this little picture again as sort of a modified version of the picture. So, I'm going to graph the function, graph the function here, this is y = f(x) and I'm looking between the region between a and b. So, here's a, and here's b, and I want to look at that area right in between there, that's my mission and somehow, that's not going to relate to the anti-derivative. Why in the world? Where is that coming out of? Where is that coming out of? Well, what I'll do is I'll take this slowly, let's take it slowly. Let me just pick a point here, x, and let's just look at the area from a up to x. So, just look at that area right there, the area that I mark in orange. I'm not going to go all the way, let's just go up to x and I'll call that area, whatever that area is, I'll call that a(x). So, a(x), just means the area under this curve from little a up to x. So, it depends on x, because if I move x, then a changes, a increases. If I shrink back x then a decreases, the area shrinks. So, this is a little machine that spits out how much area there is between a and wherever I'm located at x. Okay?
All right, now what can we say about that machine? Well, let me take a look at a low key offset of x, let me look at x + delta x, and this may sound a little teeny bit familiar if you remember way back at the very beginning of the semester where we developed a notion of instantaneous velocity, instantaneous rate of change. We took x + delta x and, in fact, that actually occurred in the definition of a derivative. But for now, for thinking purposes, let me just draw in an offset, x + delta x. This is not drawn tor scale, I want you to think of that as being extremely close to x, but if I did that, the Web would not be able to detect it. In fact, if I did it really, really closely, I try to get you to look really force... It kind of looks like this. It's sort of buffering, buffering, but that was me, that was me. It didn't freeze, but I forced you to zoom in that close. So, I understand these things are very, very close together. Well, let's see, what is this little area region right in here? Well, that little area region is going to be the change in the area. Right? If I go from here, this area, to that area, that little bit is the change in my area. So, I could right it this way, I could say that the change in the area, well, what is the area of that, well, it's approximately base times height. And the base is delta x because it's a small change in x and the height is just the value of the function, f(x). So, what I'm saying here is that as you move from this point to this point, how does the area change? How does that yellow stuff change? Well, it turns out that that area is, roughly speaking, the area of the rectangle, base times height. So, the change in area is the change in x, change in height. Okay? Imagine this is very, very tiny, because this is really pretty good approximation. If I divide both sides the delta x, I see delta a divided by delta x, is approximate equal to f(x). Now, what happens if in our minds, what we do is what that delta x shrink to zero. If I let delta x shrink to zero, what does this become? It becomes the derivative. So, what we get is, that the derivative of a with respect to x equals f(x). So, a of x is a function, I see it's derivative is f(x). How do I find a? Well, if I want to find a, and I know it's derivative, I have to anti-differentiate this. If I take the anti-derivative of this, that will give me the anti-derivative of the derivative which gives me a. So, here, I'm trying to find a, I see it's derivative is f(x), which means that a will equal the integral of f(x)dx because of this statement says that the derivative of a is f, which means the anti-derivative of F must be a. Okay, the anti-derivative of f must be a. If I integrate here, if I integrate the derivative, I just get back the function. I integrate f, I just get here. And so, the bottom line is that, in fact, there's a connection. There's a connection between taking an integral and finding an area. Finding the area.
Okay, that is sort of the math behind all we're going to do here. The math behind the fundamental theorem of calculus, and I hope by this little discussion, at least you get a rough sense that, in fact, somehow there is a connection between derivative and function, derivative of the area in function; and therefore, if a derivative equals f, that means that the anti-derivative must equal this thing a. Okay, up next, we'll take a look at a practical actual statement of the theorem and begin to see how to use it. So, we're now right on what's called the fundamental theorem of calculus, the fact that brings together the integral stuff and the derivative stuff and meshes them when looking at areas. I'll see you there, bye.

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