Intro to Integrals - The area of the wall calculation 2

Mark Misin
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34:53:37 of on-demand video • Updated October 2020

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English [Auto] Well what I can do I can't simply take a smaller interval a smaller tax I can take X equals 0.5. Right. So if I take their tax equals 0.5 then I will chop this X here into smaller pieces. So the way it would look like would be something like this. So just going to do it intuitively. Okay. Now my delta x would be this or I and this and these are the same links. So my death tax would be this and this. So you see essentially I take my Daltex and I make two delta x out of it. If I do that then what will happen here with my rectangles. Well first of all for each red rectangle now I would have two black rectangles. So I would have a rectangle here and then I would also have a rectangle here. So I'm just going to write this you see because what I do now I take this middle point from seven point five and eight. I take this middle point which is seven point seventy five. And then I go to that continuous curve and I see that aha. This is my value here. And then from seven point five to eight I have this constant value. And then I make a rectangle out of it. And then I do the same for all other red rectangles. And then I do the same for all other cases. And then you can see intuitively already that for each red rectangle I will have two black rectangles. And now my errors are smaller and I will get a more precise area. So in this case then I do it. I'm not going to do it for everything but I'm going to do it for one more. So I'm going to take this part here and I'm going to go to this black line and then OK this would be one of my levels. And then here from 6 to 6.5 I take the middle point six point twenty five. I go to this curve here. To this black curve. And then from 6 to 6.5 I will hold this value constant. And now I will have two rectangles two black rectangles instead of one Red Rectangle you see. So now I will have more rectangles and I can do it for all red rectangles. So in total I will have 16 rectangles. Now if I want to move away mathematically what does it mean. It means that this term here does X become smaller. Right. But then the amount of wise will increase. So I will have an area that I can approximate. And in this case now it would be now delta x I'm just going to put it small. Smaller and then here I would have to buy one plus why two plus why three plus up until plus. Why. 16 k or the summation sign would be I or delta x small equals 1 2 and equal 16. And then why I so you see on one hand these daily tasks become smaller. Here my Daltex was one. And now here it's 0.5. So my Dell takes decreases but the amount of Y's increase and now I have a better approximation of my area. Now what happens if I decrease my Daltex even further. Well I will simply have more rectangles my death will be smaller but I will have more whys and if I continue doing that and I say that my delta x approaches zero then the amount of wise will approach infinity right. If my Daltex approaches zero then the amount of these terms will approach infinity and in the end I will get my exact area because remember right now with rectangles I am in my discrete world. Right. So right now as I approximate my area I'm getting in discrete area but let's say that the most precise area would be a continuous area so a continuous area I get if I work with a continuous function and this problem here it's a continuous parabola. So in the end I mean interested in knowing the exact area under this black parabolic curve. So I need to get this continuous area but I have been able to approximate that continuous area with with rectangles or with the discrete area. But now if I say that my interval my delta x approaches zero then consequently the amount of Y's will approach infinity and then this entire multiplication this entire product in which one term approaches zero and the other term approaches infinity this entire product will ultimately resemble into my exact area. In other words my discrete area will approach my continuous area OK. And here I want to note that when I say that my Daltex approaches zero it's not the same like saying delta x equals zero. It's not the same. These are two different concepts. Right. When my delta x equals zero then obviously my term here would be zero. And then I shouldn't have any area at all because everything multiplied by zero would be easier. But in this case if my Daltex were to be zero then my then the amount of Y's here would be infinity. So there some would be infinity. So I would have some kind of multiplication of zero times infinity. Right. So this will give me infinity and this will give me 0 and that kind of thing is just undefined. Mathematically it's impossible to define it so it's undefined meaning you don't know what you will get here could be eight fifty two million. You don't know you just can't determine what this product would be if you were to multiply the zero times infinity. And that's why this equality sign right this belongs to algebra and algebra is too weak to solve a problem like this. And that's why calculus was created and calculus came up with a new concept instead of equality. I'm going to say that something approaches zero. So when the text approaches zero then I can see what my area will approach if Daltex approaches zero then well then of course this term will approach infinity. And then I can use the limits that we have covered in the past limits to find what this total equation. This entire product. What it will approach and remember it will approach something. It will approach your exact area. It will not equal it will approach your exact area. So your exact area would be like a theoretical limit. In other words you can think of it like this very simply as Dale thinks gets closer and closer to zero right then your discrete area gets closer and closer to a continuous area. You can always get closer to zero. You can always have zero point one and then you can have zero point zero or one and zero point zero zero one. So you can always get closer to zero. And the way you should think of it is like this. The closer you get to zero the closer your discrete area approaches your continuous area. In other words the errors that you get here they get smaller and smaller and smaller. So as your dealt tax approaches zero the error the errors also approach zero and you're discrete area approaches continuous area. And now when that happens then you're your some because essentially it's the sun right. The other name for this is some because you multiply something then you multiply something else and you multiply something else. And then you have a sum. And that's why you have this summation sign here and now this summation sign is a discrete summation sign. Discrete summation sign OK. And when you're dealt Thanks approaches zero then you're discrete some approaches continue some and continue some. We have a different sign. If this is the discrete summation sign then when you're discrete some approaches your continues some then this sign here which is for discrete summation will become this sign and this sign is a continuous summation. So you use this sign when you're dealt thanks. Approaches zero and when the amount of Y's will approach infinity and when you go from your discrete world to your continuous world then you have this continuous summation sign which is called integrals. And there you have it one way to look at integrals is that it's a continuous summation. All right. This signing here is discrete summation. You're summing some kind of blocks. But as the width of those blocks approach zero. Then as you go from your discrete work to your continuous world then this summation becomes an integral and therefore if you now apply the concept of integral to this function here you will calculate the exact area that technique will allow you to calculate the exact area of this wall. And in the next video I will show you the integral from a different point of view. And I will also dive into the math of integrals so that you could actually mathematically calculate the area of this wall. The area under this black curve so that would be the objective for the next video. Now you know that if I take my function which is this one. Y equals 16 minus. And then parenthesis X minus 4 and a square. If I take the integral of it right then I will get the exact area of the wall and in the next few years I will show you how you can do that mathematically. So see you in the next video.