Calculus: Approximating Areas of Plane Regions

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The Basics of Integration
The Fundamental Theorem of Calculus
Approximating Areas of Plane Regions Page [1 of 1]
Well here we are at the last major issue I want to share with you. And it really comes back to the very first questions that I posed at the very beginning of our discussions together. And that was really the two basic questions of calculus. So what were those two basic questions of calculus? Well, the first question was how do you find instantaneous velocity? And we actually went off and spent an awful lot of time working on that and thinking about that, and we saw that the answer was, well, what we ended up calling the derivative. The derivative empowered us with the ability to find instantaneous rates of change. And we looked at all sorts of examples, whether they be slopes of tangents and the math verification of those to graphing, whether they be instantaneous velocity, instantaneous rate, and looking at applications to relay rate and other things. We saw a lot of examples of that kind. Now what was the second question I posed? Well the second question I posed had nothing to do with rate. Had nothing to do with velocity. Had nothing to do with any of that stuff. Instead, it was a geometrical question about area. How do you find the area of this rectangle? Well it's not that bad of a deal. What would you do? You would just take base x height. The rectangle is such a simple shape, that to find the area of it is actually not a big deal. Okay, but what if I actually made the shape a little more exotic. Suppose I took some scissors and did something like this. Look at this. Do you see that? Now I'm taking what was this very simple thing, a rectangle, and I'm making it now into a much more exotic shape. Look at that. How would you find the area of that now? Well see there's no obvious or easy formula for the area of this thing. It's certainly not base x height, because the top is all wiggly and stuff. And this was the second question I posed. How do we find areas of really weird shapes? And then, in higher dimensions, how can we find the volume of really weird shapes, like a donut and what not? Well, that's an interesting question. And that's a question that actually has a lot of applications, whether you're actually building things, or whether you're just thinking about how things work or how large things have to be. And notice this question has absolutely nothing to do with notions of tangent lines and instantaneous rate. But this is the second question of calculus, and this is the question that I want us to take up and examine now. Namely, how do we find areas of very exotic shapes?
And let me try to solidify this, and in fact I think this example that I just made here, captures the spirit of what I want to look at. But in particular, let's suppose I have a graph of a function. For example, suppose I have this graph y = f(x), that's some function. And I graph it, and we can do that now because we know how to graph functions. We can find critical points, max/min, increasing, decreasing, concavity, that's another story that we already did. We have this nice graph. Now suppose I want to find out, if I fixed two points a and b, and draw these lines, I want to find out what is the area between or under this curve and above the floor, the area between the curve and the floor. That's some shape and, in fact, depending on what the shape looks like. And the question is what is the area of that yellow? That was sort of what I inspired here by cutting this out. You could think of the floor here as the x-axis and the top here as some exotic function. And the question is how can we find areas of these kind of regions? Well it turns out that, at the moment, it seems like an extremely hard problem. So what do you do when you're faced with a hard problem? You don't do it. That is a great thing. When faced with a hard problem, don't do it, you can't do it. Instead, try to do an easier problem that might inspire some idea or some insight that might lead to a solution to the more difficult problem. So how would you find the area of some sort of exotic shape like this? And we don't know a formula for it so it seems pretty hopeless. So let's try to reduce it to an easier problem. Well an easier problem was the problem when I just had a rectangle. Remember I originally started off with a rectangle. Well that was the problem that seemed so great and so easy that it's worth thinking about at least for a second, because what could we possibly do. Let me draw now for you an example of a function. Let's put in this function right here. So here's a function y = F(x). And what I want to do is that I give you sort of two endpoints here, and I think we called them before in that last picture, I called this one a and this one b. And the goal, the really hard problem, which we don't know how to do, is to find the area under that graph, but above the floor. So this is the question, how do we find the area of all that? And the answer is we don't know. I don't know how to do that.
So instead let's do a problem that we can do. And what we did, at least with the derivative stuff that we developed, was we said, "Well if we can't do the problem exactly, can we at least approximate it? Can we at least make a guess? Can we at least estimate it?" Well I think we can estimate this by using shapes that we know the areas of. And one thing I know the area of is the area of a rectangle. That's easy, base x height. We already saw that. So what if I actually put some rectangles in here. Let me fill this thing up with a couple of rectangles. For example, suppose I put a rectangle, cut this right and half let's say, cut it right in half and put a rectangle right in here. And then put a rectangle, let's say, right in here. So I've got this rectangle here and this rectangle here. Well if I find the area of that rectangle, which is pretty easy, and the area of this rectangle and add them up, I get an approximation to the area of the curvy thing. Not a particularly good one, but I do get an approximation. Now how could I get a better approximation? Well one way to get a better approximation would be to maybe cut up the base of the rectangle and make them a little smaller. So what if I cut up the rectangle, this region, and make four rectangles. Then make one rectangle like this and one rectangle like this, one rectangle like this, and one rectangle like that. We want to find the area of that rectangle, and add it to the area of that rectangle, and add it the area of that rectangle, and add to the area of that rectangle, well that's going to give me actually an answer that's a little bit closer to the actual area, but still not exact. But you have to admit it's a better approximation than the first one I had with two rectangles. So if I need a closer approximation, what could I do? Well I could actually make a lot of rectangles in here. When I make a lot of rectangles in here, you see then each of the rectangles themselves, the area of those things not a big deal, it's just base x height. And so we can compute that so it's not that difficult. And yet, when I add up all those areas of all those rectangles, I get an extremely close approximation to the actual exotic area that I'm looking for. You see that? I have these rectangles. So what I'm doing is I'm taking one hard problem, and I'm converting it into a lot of tiny, tiny, tiny problems, but each problem is easy, because each area is just base x height, not a big deal. But if I do a lot of easy problems, that becomes equivalent to a very, very hard problem. So the more rectangles I put in, the closer I get. How many rectangles do I need to have in there in order to get the exact answer? Well, I've got to make those bases and smaller and smaller, because once I have the base a certain size, there will be a little error. There's a little error up there. So I've got to make it even smaller, and smaller, and smaller so the bases actually have to sort of approach 0. We have to sort of take a limit as the bases approach 0, and by doing so we're putting more and more rectangles in there. And so, in some sense, if we take the limit, we're actually putting in infinitely many rectangles and stacking them up. And then I get the exact area, if I put in infinitely many rectangles, where all the bases are essentially 0. So you can see already there's sort of calculus spin here. Somehow, I want to let things go to 0. I'm putting in infinitely many. I want to take a limit. So there is something here, but the question still remains, how can we actually find that quantity?
Okay, well up next, we're going to try to address that issue. Take a close look at this. In the next lecture, I'm actually going to try to inspire the big answer to come. And the big answer to come, by the way, is a really big theorem. In fact, it's known in the math world as the fundamental theorem of calculus. So this is it. This is the big time. We're approaching it. So up next, I'm going to try to inspire it with a little bit of advanced math so you can see where this thing is coming from. See it coming, and then we're going to get it. Okay. We'll see you there.