Alright, so the idea is this, right? You have some curve, let's say tread Hawaii clothes Fx, OK. And another color it green, y, g, x. And then you have two line, X equals a and x equals b. Ok, and you're interested in finding the area of this region. Okay, So let's call it the PQ and RS, these four points, right? So we talked about last time that if you want to find the area of the region PQRS, then So what I'm going to find the area, let's call this a and B. You find the area P a BBQ, and subtract from it our abs. So goal is find the area of the region. Pr as q. Ok. This is the region between two y equals f x. That's the top curve, y equals g x. And the lines x equals a, x equals b. Alright, that's the area we want to find and seen that area. Pr SQ is the area of the big area. P a B Q minus area, or a BS. Is this okay with everyone what I'm saying, that this big area, the area under the P a, B Q is sorry, area P R S, q is the big area minus the smaller area. Okay, with us. Tailor that, okay? Okay. All right. Now the big area, P a B, Q is the area under f. And subtract from it the area under G, right? So this big area is the area under the red curve. And we know that the integral and this secondary, the area under the green curve, so area under F is integral f x dx. Okay? That's that the area under the green curve. This area that is area under T. Okay? And we can combine this into one expression. So it's the area f x minus g x. This integral. Ok. So this is the final result. I'm going to write it another way, but this is it. So therefore, area PR is q is the difference of this things here, okay, I want to point out something here because that's how I'm going to write it. So the area PR SQ is integral of f x minus g x. Now f x is the top curve and g x is the Baltimore upper curve and the lower curve. So this is, I'm going to write it, okay? So we're going to call it a proposition. Sketch. P or Q is equal to integral minus g x dx. Okay? Now this f x g x is not easy to remember, so it's better to I remember it like this. So it's y upper minus Y lower dx. Okay? That's easy to remember. F t, you don't know which they are. So it's the upper curve minus the lower curve. Okay? And the integral from the interval a, b. So this is it. All the problems in this section. I just wanted to come from this. We will apply it and I wouldn't want to call it clubbed very clever, that nothing very clever here. And what we're going to do but slightly different ways, just this very formula. I have two curves, y equals f x, y, g x. And this is the area above lines x equal to x, sorry, between X equals a and x equals b. And the area between them as integral a to b. Take the upper function and subtract from it the lower functions. That's it. Alright, so let's do the examples now, some examples. Ok, why upper minus Y lower? That's how I remember it. So we'll do the first example is the one we wrote down last time. And all these questions, I will give you the schedule because we haven't spend so much time sketching curve. So I'll give you the sketch. The area of this region. Integral y upper minus lower dx. And what are the limits a and b for this hexes between its pretty obvious, right? 0 to two. So x is equal to two. And that's it. That's the idea. Okay? So 0 to two, y is four. Y lower is 3x minus x squared dx. And we know how to integrate. We've done this many times. I'll do this quickly. 4x minus 3x squared over two, x cubed over 30 to 40 to two. So that's minus. And when you plug in 0, you get 0 for everything. So this is eight minus 62614 over three. Okay? The integration is pretty straightforward in this case. Okay, let's go back. Hence the integration part. I think you've been doing a lot of these problems, so that should not be an issue. Here is the main point. You have a sketch. You have the upper and the lower curves. You have the a and b X equals a x and b and the formula for the HIV, Yeah. All right, so this is the most basic application. Now let's look at variations of this. Find the area of the shaded region in the sketch. So I'll read. And the curve y equals 2x plus one. The other one is y equals x squared minus four. The shaded region. Ok, now in this problem, I don't think so for this region, what is the upper curve? So you have to focus only on the region. Why? Do you all see that the green curve is the upper and the lower curve is x squared minus four. That's no doubt. Great. Question is, what does the a and the B? What are the x limits? How do we find them? I think we all agree that the x limits are the coordinates of these two points, right? The x coordinates of these two points. That's the x limit. Let's call them a and B. So the x limits will come from the x coordinates of those two points, u naught, so you will have. So now the question is, how do I get that? Ok, so let me just write what I've said so far. Ok? So y is 2x plus one minus y lowered as keep your parentheses otherwise, you know. Okay. When the question is, what are these two things? Okay, let me write them as a and b again. So a comma we will be obtained from D. So what is special about it and be like in the sense that how would you okay. Yes, we have the question, How would you get a and b? So I need the coordinates of capital. I need the coordinates of capital a and capital B. Uppercase a and B. Can you recommend a strategy? What is special about a and B? Like what distinguished from any other point in the x-y plane where the curves intersect, that is eight, right? So we have to find points of intersection from the points of intersection of the curves. Who, why? What was it? 2x minus one. And y equals x squared minus four. Points of intersection means I'm looking for a point which lies on the green line as well as on the blue parabola. So I'm looking for a point, points x and y, which satisfy this equation and this equation. Okay, so we're looking for solutions of two equations in two unknowns. That's what we're looking for. An intersection means that lies on both curves, right? So he points x comma y, which lie on both curves, right? So x, y. So this is true. And this is true for the points of intersection. So how do I find the x and y which satisfy both equations? Normally used for solving one equation, one variable. But now I have two equations and two variables. So I'm looking for x and y, which will be this equation and the same x, y will be this equation. Same XY. Any recommendation? Okay. So that's a little big, but tell me what do you mean by set them equal to each other? For money, move to x equals 0. Exactly. Perfect, right? The thing is the same. X and y obeys this equation than that. So that y is 2x minus one and that y is also a x squared minus four. That y is 2x minus one, and that twice 2x squared minus 4x. So therefore these two must be the same. Let me write it another way. Another, ok. So that is true. Now we have a single equation in one variable. So you bring everything to the X squared site. Good thought. You can factor it. You get x minus 3x plus one is 0. Okay? And once you have X, you can get why. You can use either equation to get the y. Now that you needed actually, y is minus three. Okay? So then you can go and so this is minus one and minus three. That's 35. Right? So that comes later. But, but this'll be important pigs. So these too, so we have done, is I can also figure out the points of intersection of the two curves. You have to do this. To x minus one, y is x squared minus four. These are equal. Make it into a single equation in one variable. Alright? So now we can finish the problem. So x goes from minus one to three. And the other part we've already done, it's 2x plus one minus x squared minus four. So minus one to three, 2x minus mc squared plus dx squared minus x cubed over three plus 3X. Okay? And I can finish, it turns out to be 32 over three. Okay, so the same idea that y minus y lowered x a to b, just used slightly differently here. Just slightly differently. I needed to figure out the points of intersection. But otherwise it's the same idea. Now let's do another variation of this idea. The goal is so pretty straightforward. What's y upper and lower? So let's start with what is y upper for this region? Jodi wanted to recommend something equal to each other. So we have to do the y upper minus lower. Yes. So what's the upper curve for the region x cubed. Okay. Is that care or if there's some issue with that, is that all of us the upper curve? Yes. What about here in this area? We don't know that one. So he's at the upper curve with a lower curve there. Lower, right? Yeah. So here y equals x cubed is the upper curve, but here it is the lower curve. What is the job? What does the lower curve in this region? Let's call it region one, and let's call this region two. Region one and region two. So Joe, can you tell me what's the lower curve in region one for region one? Or is that still x cubed? That is the upper curve, right? So what's below the lower part of that region? Two, we now have that equation. Can you guess what it will be or can you think of what the equation would be? Can you describe the shape of the Lord come what is the lower curve? On? Its like? A parabola is a curved note. So for this region one, what is the lower curve? The top curve is the green one, sorry. Right top curve with the green one. What is the color of the lower curve? To the green one? Is the top curve, right? That is the upper curve, sorry. So what does the lower curve for one to one or negative one? It's occur, right? It's going to be something like this or like this or whatever. So for the region one here, what's the lower curve? Sorry, Joey said something. I'm not sure. Okay. No problem. Y equals 0. It's the x-axis, right? The lower curve is the x axis or in region one. The next Central. Yeah. Okay. And then the x-axis, the equation of that curve is y equal to 0. So in region one, it's y equals x cubed is the upper curve and y equal to 0 is the lower curve. Okay? But that is not true in region two. So first thing right away, you see that I need to break it up into two problems. I need to do is area of region one separately, an area of region two separately. Okay. And for the region two, what is the upper curve? Alexander, do you want to say something? What's the upper curve for region two? I'm not quite sure. Okay. You don't have to give me the equation, but can you see the shape which is on the upper curve for that region? Is it the y equals x cubed? But that's the lower part of the curve, right? That's below the region. Right? These are the top curve and that's the bottom curve, right? So what does the equation of the top curve? You see that it's y equals x axis. Right? So that's the point. So okay, for one, for one. So for area one, the upper curve, the green curve, and the lower curve with the X axis. Okay, fought to upper curve is the x axis. That is y equal to 0, and the lower curve is the green curve. Have you on an agreement about this mix hint. Alright, so let's do the cave. So it has a CO2 here, your one plus area a two. Okay, so Y minus Y lower. So area one. Why Upper was here your one is x cubed, dx. Okay. What are the limits for a and b for the upper region? Just 01. Okay. And what are the a and b for the region to Jessica, you want to tell me 0 and negative one or to put it like you always put the smaller 1 first, so it's negative 10. But that's correct what you said. So let us do that. So for the upper one it was 0 to one. And if we do all of this will be one over 44 area to a to b y upper minus y dx. Okay? And it goes from minus one to 0. Upper curve is the x axis, the lower curve is the green curve. Do all that, you'll find that's also 104. So Aereo shaded region is one over four plus one over four. That's off. Okay. Let me show you another variation of I'm not going to finish it today, but let us talk about what are the issues here. It's going to be again, the same idea, upper minus lower H20. That's all we're going to use. You just have to adapt to the situation. Okay. Okay, so I have this region formed by three curves. What I'm looking for is the area of this region. Okay, we'll talk about that and then we'll finish this problem later on. So for this region, what's the lower curve? The curve, it is always at the bottom of the region. The bottom boundary, if you like, n squared, y equals x squared, no doubt, right? What's the upper curve? Iceland? You want to say something about this? Why? Because I'd x-ray to have innovative. So is that all of us, the upper boundary of the region. Now, you like to have their badge is right. So if you have to do this problem, we should break it up into two parts and correct, because the upper curve switches from one part to the other. Where does the good Where do I separate holly? Team of which two curve and the green and the pink one, right? So I need to cut it there. What color can I choose? So I need to cut it here. So I will do this region, region one. Let's use a different color. Continues this. So that is region one and region two. I need to break it up like that. Okay, then in region one, upper and lower curve determined region two. So I need to figure out these points of intersection. And then we'll be able to finish the problem. Okay? So it's the same idea. You just have to adapt it to different situations. Alright, so I'll do this next time. And so the plan is that I launched a homework questions and we'll
Area between two curves I (6.6), Math 221 Winter 2021
From Rakesh Rakesh February 03, 2021
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