Gentlemen, this is a problem. Number 43 from section 5.2. And essentially, the problem is discussing reminds sums, and we want to over some interval, um, forgiven function. And we want Thio. Find a formula for the and three mon some and then take the limit. Okay, so our, um our function here today is the, um is equal to f of X R f of X equals to one minus x squared. Okay. And now, to give you a brief idea of, but we have to do our, um Let's take a look at the graph here, so I'll draw the, um uh, so I'm gonna draw the grab ups. Come on. Okay. So I'm gonna draw the graph of the function here. Looks like it looks like this. Um, so we'll take the interval like this. Um, and, uh, course, this is, um are interval down here from 0 to 1. Um, and our function looks roughly like, um, this here. Okay. Um, no. So what we want to do is we break the on the end. Um, on the end, some Ah, we break the interval into end equivalent, um, sections or ah, absolve the intervals of length one over end eso. In this case, it goes something like this. So, for instance, if we're, um, looking at, ah, the 4th 1 are made Maybe the 5th 1 here, we break this into five different subsets. So this is, um this is won over five hoops, so this is sorry about that. Okay, so we have, um um one fist. Um, this is a ah, boy. Come on. Okay. Um uh, this is two fists, two fists and so on street office and four fists that's so on and so forth. So this is, uh this saree here is, uh, 4/5. And, um, this one in the middle is street office. And, um, then the, uh So you mean, uh and then, um, at each, uh, beach endpoint. What we're doing is we're forming. Um, we're forming a, uh ah. So you you form a box here, Basically a rectangle. Um, and, uh, this here we form rectangles with this, like this, and this s so we're using what they call the right hand endpoint for the boxes. So that means that the height, if you will so are with, of course, is the, um are with here is the 1/5 or with here, of course, is going to be the one over. And, um, and our height here then will be, uh, this This our height, if you will, is, um the, uh, ass off the height is half of in this case, it would be half up 2/5. So it would be our height would be s of, uh, to this. So when you calculate the area of this box so the air will call this box, maybe call the box? Uh, A In this case, it would be, I guess, a two. Okay, So this box here well, just name it as a too. Okay. Now a to, um Izzy s Sorry. The area. Um, yeah. Maybe I should have called it are two instead of a two. Yes, because our two would have been a better letter, I guess Are, too. So the area of our two from the area of and chu, um, is equal to Well, it is half of 2/5. Um, to this, uh, multiplied by the, uh, the with, um, which, of course, is one over. Yeah, and another way, which is kind of the ways that we're using for this, um is the the way that they use in our the way that the reminds some set it up is that this is equal to, um this is equal to, uh yeah, that's ah f x will sit roll call it F k um, times the delta dull Tahir uh, x k and what? Delta what this is referring to. So the delta of X k but it's actually talking about is what is called the increment, um, lengths. So if you notice so in particular here, the increment length is the is the difference between the two end points. So in particular it's, um it would be, in this case, two fists minus 1/5 or otherwise x k minus x k plus one or something like that. And in particular here, reusing the right hand and points. So the right hand endpoints refer to the end point on, I guess the quarter right hand side course, if we're using the left and end point is then, um, the boxes would look a little. The rectangles would look a little different in that instead of the one we have. It would go, Uh, they're rectangles would look like So. So the the area of the given rectangles would be a little different. In that case, obviously, they would be larger and here. But, um, the the idea, at least in these remind sums, is depending on the point to you two, you're going to get a different areas for each some. And, ah, then the, um is the ends Riemann some is given by or these two formula that they that they use in the book looks like this. Um, it's ah ass And, um is equal to, um, the Sigma notation, of course. Ah. Which, um, you know, looks, uh, like this guy. So, um, as, uh, yeah, says, um, Sigma of half of X. Okay, um, times the as I say the delta the dealt er excuse me, Delta X, k and crucially, here, um, the ah, end this and be in this problem. Um, were our increments are gonna be of the same length. So in each case in this problem, um, the Delta X case will, um, be equal to one over end. Okay, So are you on this problem anyhow? And other problems will be a little different, but in this problem, the increments will always be of length one over and And the, um, f x case are always found by the right end point. So, um, are are, But we have to do is first find a closed foot what they call a closed form or s in here, and then use that to calculate the limit. So, um, with this in mind and let's begin. So, uh, first off, um, the ah for you given ends of just an arbitrary number. And, um, we get that r s in here, uh, our s and, um, again the, um Oh, the increments. So since we started, So So it starts at, um, Kay. Since we're using this point are for Kes. Our first rectangle will go from if we replace instead of five here. If we replace this by end, Goethe one over and you over. And, um, you know, dot, dot dot This would be maybe instead of three over. And, um, this would be ah, Villa A. This would be at three over, and and then, you know, one, of course, is is is, um is an over end. Okay, so the one before it would be Ah, um ah. Let's just say how do you want asses and minus one over and yeah, it's getting a little tight here, Um, but and minus one over in and ah, and minus two over and all the way down to ah, and then three over and two over. And so B s. And, um So the first thing we should try and figure out is what is our, um what does our index k go over eso in this case, Kay goes from, uh, que goes from one because we're using this side, remember? So it's k equals toe one. Um, k equals to 12 And like I said, uh, it ends at one. So in this case, it's K equals one, two, um, in here So it's K equals one to end. Um, and then the f x k refers to the this right hand endpoint. So that would be, um it would be f off. It would be af off, uh, K over end and then times the length, which, of course, is, uh, one over. And so I'm gonna simplify this now just to eliminate it like this I'm gonna just write it as af of K over and, um, divided by Yes. Okay. It's simple. Find that here. Ah. And now, in this case, of course, the function I'm using is one minus x over two. So I'm just going to replace af of axe over and buy one minus X over too. Um so let me just do that here in this form. You here. So I'm just gonna replace this by, um, this becomes, uh, one minus. Uh, Now it's k over. And so it's, uh, que over and square. Okay. And, uh, okay. And now we want to do, of course, is continue to simplify this now. Ah, so you'll notice that your summing from one end. So if you're something, um, from want and ones, then you're going to get in there. But then you're dividing by ends. You're going to get one. So this is equal to, um this is equal to one and then minus um uh, is this some? That's some again here. So it's going to be minus whips. Uh, Minus again? Yeah. Is the, uh, um K squared? Uh, we're gonna get troops, huh? K squared now. Provided by and square times end gives us it's divided by n to the herd. Um, provided by, uh, case by the end square times end, which is one over our wishes and cube. And again, Now, the sum goes from K equals one. Uh, so this some goes from I want to end here. So this is our some and, um, Eagles want to end. So now we're now we've calculated Are we calculate ups. Okay, so we've calculated this is ah, um, this is our least our first form. You. LF you Well, this is our first formula for for, uh and you're sorry for s And so this will label as maybe, uh, f one. Okay, so we'll call this F one. Okay, so now with f one in mind, um, now we have to move forward here. Uh, we can't. We need to use a formula. There's a nice formula to make this easier. So So we moved to the next stage, and, um, we want to find a formula for the end. Okay? Eso We use the formulas that, uh, this guy here, which is, um uh okay, So we used a this formula, which is actually in your book, So, um, but I'm just repeating it Here, Um, this formula here it says that the squares So the, um a equals one. Ah, to end here. Ah, wake up. Come on. Sometimes it's gonna be so frustrating. Okay. S O okay. I'm sorry. I see what the problem is. It's too close to the top here. So in other words, it goes from K equals one hand. I'm sorry. It's just too close to the top. K equals one end of this guy is equal to this end and plus one time to n plus 1/6. So we use that. So then, um, from F one. So from f one, we get, um, from F one, we get that, uh, s and is an equal to one. Um, minus. Now, remember, we have. So, uh, we have this, um uh, the the F one. Oops. Uh, So the F one was this case, This some divided by n cubed. So maybe maybe what I should do is actually factor out this and cube here. Ah, see? So if I factor out the end cube, uh, you'll see that this some here is actually the, uh, and shoot. Um, so this sum and now it's one minus the sky. So it's so this here we have from, uh, this This formula here gives us one minus. So, um, it's one minus this. See, in n plus one, sometimes two and plus one. Ah. And now it's divided by the six times the end cube from the formula week from what we had before. So this is then six thistles. Six and to the third. Okay. Ah. And now, um, what we can do is we can expand this out. Well, actually, I prefer to just cut to just cancel the end here, and, um And that would make the denominator ace, too. So instead of that, I get and squared. Um, And now what I get is that this is then equal to one. Um, minus. I'm gonna write out school. I'm gonna use the foil method to compute the numerator of that fraction there. So I get, um, to and squared, uh, plus three and plus one. And, of course, it's now divided by, uh, six and square. So we get here, um, six and square, dividing each, uh, each fraction. Uh, I'll be just sort of standing the given fractions here. Uh, me. And so what does this, Uh, what does this tell us? Well, we have then is ah, we can go through then. And in particular, I want to simplify this down, so you'll notice that the end squares cancel in this 1st 1 So I lose the end squares here, and this becomes one over free in the 2nd 1 The end cancels with this one, and I get Well, it doesn't matter here, but the final one is just 1/6 and square, so I can simplify this down further. One minus 1/3. Ah. Then gives me, uh Well, let's let's just move on to the next one. So So one minus 1/3 would give me 2/3 here and then plus this of their stuff. So, in other words, I get, um that my s and by s and eyes equal to, um you see, it's equal to 2/3. Two over three. Plus, these other guys, which are, um, or I should say, minus one over, uh, to end. And then it should be minus. Sorry. Um um, minus one over to end and then, um, minus 1/6 n squared 1/6 and squared. And so this is the formula for my s end for my aunt. Uh, sorry. My ends reminds some. OK, And now what I want to do is just compute the limits. In other words. Now what I want to do is just look at the limit. Um, as an approaches infinity, um, of s and and, of course, uh, this is, uh, is equal to, um the limit is end goes to infinity of that, you know, some above, which is to over three minus one over to end. Um, minus one over six and two. The second. Ah. And now, hopefully, you know your limits Well enough to know that this these two terms here, both of these two terms are gonna go to zero. So this is going to go to zero. And this will also go to zero. And what we're end up. What we end up with is then, um, just to third. So we end up with 2/3, and that is our answer. Um, eyes 2/3. And that's it for Ah, this problem, folks, it's a bit of a long, long problem. It's, ah, number of steps, but, uh, that's it. Uh, thank you very much, folks. And have a nice day.