Evaluate the upper and lower sums for $ f(x) = 1 + x^2 $, $ -1 \le x \le 1 $, with $ n $ = 3 and 4. Illustrate with diagrams like Figure 14.
Yeah, yeah. So problem 5.8. We're dealing with the area under curves. We're asked to find the upper and lower sums as estimates for the area For the number of rectangles being three and 4. So real roughly this is a parameter that opens upward. So think about this And it's been shifted vertically up one. So the graph of this guy, it's going to look like this. And so we're interested in finding estimates for this area that you see that's all shaded. Okay, So what I notice is the the curve is decreasing from negative 1-0 And it's increasing from 0 to 1. So that means it's going to be a difference. In terms of when I look at upper sums versus lower sums, I'm going to have to shift which side of the rectangle I'm using for the heights in order to make that happen. So that's the rough curve. Let's look at something a little bit more precise. So if I look at this number of intervals equal three. So I'm going from negative 1 to 1 on the X axis. And if I use three rectangles, what you can see is if I start out, if I use the left side as the height then that first rectangle that's an upper some. And if I look at the next one, if I use it's gonna be it's gonna be either way the left or the right, I still get an upper because the rectangle is above the curve. And then you can see that. Well that last section there in the last rectangle I cannot use the left some I'm going to have to use the right. Some Give me the right amid the right end point of the rectangle in order to make that an upper some. So if I'm using in equal three Then how is this going to work? So the distance from negative 1-1. So the width of the rectangle Is going to be 1 -2, divide that by three. So that is 2/3 is the width of the rectangle. So for the end. So let's look at the cases that we have We're gonna have in equal three And in equal four were asked to come up with an upper some and a lower some. Yeah So to find the upper some when in is equal to three. So you're gonna see it's going to be um 2/3 is the width of each um rectangle. So 2/3. And then that first rectangle is going to be the value of the function and negative one. So this is going to be two thirds. Yeah f of negative one. Yeah. Plus and then the next rectangle is going to be, it doesn't matter. It's since this is an even function F of negative one third or f of one third. So plus F of one third. And then that last rectangle instead of using a left endpoint, it's gonna have to be the right in point. So F. Of one. Yeah. And if we evaluate this ah what do we come up with F of negative one is two. So this number is to um and then also this number right here is to F. Of one third. That's 1/9 plus one. That is going to be 10 nineths. And so this is going to be uh if I look at that real quick so four plus 19th. Um So I think that's four 30 46 nights. So this is 2/3. Yeah. Times 46/9 Which is 27. Yeah. Over 92. Yes 92. Yeah so that's my upper some using three. Um Now if you look at the lower some go back to the curve here in order to get a lower someone should change which side of the rectangle that I'm looking at. So if I look at the right side you can see by looking at the right side um that first rectangle will give me um a lower some. The second rectangle. I can't really look at the left or the right. I'm going to need to look at the mid point to make that happen And to get a lower some in the last case I can't look at the right. I'm going to have to look at the left side. So my points are going to be negative one third, zero and 1/3. So now for the lower some It's going to be 2/3. Mhm. Just look at that one more time to get the lower some is going to be f at negative 1/3. Yeah. Plus that second rectangle to make that a lower some. I'm going to need to use the midpoint. That's the only the point on the curve that I can get the lower side. That's gonna be F. Zero. Yeah. And then to get a lower some on the last rectangle instead of using the right side I'm gonna need to use the left side which is F. At one third justice. Mhm. Okay. And you could do that out on your calculator now to go to end equal four. Let's go back to our calculator Lest adjusted to in equal four. So 4 rectangles. Ah It also helps if I adjust my scaling here to just step it by And by 1 4th. Yeah. Mhm. And that would be good. So now if I look at this in order to get an upper some. So in order to get an upper some it looks like for the first two points I'm using left side rectangles to get upper psalms. For the second I'm going to use the right side because it switches from decreasing to increasing their. So now the distance that whole distance from negative one is two, Divide that into four intervals each. Each rectangle is a with 1/2. So now I get this is going to be one half. Yeah And if you look at this it's going to be f at negative one And then f at negative one half. Yeah. Plus the second two rectangles instead of using left in point going to have to use the right in point. So one half and one. Yeah. Yeah. Yeah. So that's going to be my upper some to find my lower some in this same case let's just switch instead of using the left in points and to use the right and you can see that. Okay. Yeah. For the first part of this curve, the first two rectangles I'm getting a lower some by using the right side of the rectangle. Well I'm going to need to use the left side of the rectangle for the second two rectangles to make that work. So this is going to be for the lower some it is 1/4 and again you're using the right side so F of negative one half and zero. Yeah. Yeah. Yeah. For the second two rectangles instead of using the right side, I'm going to need to use the left side so zero and one half. Yeah. And that's how I get the lower some in the case of in equal four