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Evaluate the upper and lower sums for $ f(x) = 2 + \sin x $, $ 0 \le x \le \pi $, with $ n $ = 2, 4, and 8. Illustrate with diagrams like Figure 14.

$\mathrm{upper}$ $(f(\frac\pi{4})+(f(\frac\pi{2})+(f(\frac\pi{2})+(f(^3\frac\pi{4}))\frac\pi{4}$

$\mathrm{lower}$ $(f({0})+(f(\frac\pi{4})+(f(^3\frac\pi{4}))+(f(\frac\pi{2})+(\pi)\frac\pi{4}$

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Yeah. So problem 5.7 were given a function here, F of X is too plus the sine of X. So, you know, just by looking at that, if we're also looking on the inter role from um X is from 0 to Pi, you should know that as a sign curve that's just been shifted up to. So if you look at that is going to start at zero, here's Pie. So the curve is going to start here in here And then it's going to go up to three. So we look at this, this curve goes up and then it comes down to one and are right there and sorry about that, back up. I did it, that was to two pi so if it goes to pie, sorry about that, it's going to start at zero and 20 and it goes up. Yeah, Value to three or right here. So what we're seeing is just one one half of a cycle of the sign curve. Okay, so that's what's gonna look like they want us to figure out for different values of N. N. Is the number of rectangles that I break this up. What would be the upper some and the lower some. So let's just take a look at this. First of all we want to look at when N. Is equal to two. So if N is equal to two, I have two rectangles, how do I get the upper some and the lower some. Okay. So what I'm looking at now is the right in point was used to determine the height of the rectangle. So you can see the curve is increasing up to pie or two decreasing from pi over 22 pi. So yeah, the right in point on that first rectangle gives me an upper some in order to get an upper some on the second rectangle. Instead of using the right in point, I'm going to need to use the left end point to make that happen. So it looks like I'm gonna have to use high or two in both cases as the height of the rectangle. So in this case if I do the estimate, uh let's go back to that. And so if I look at that, the estimate of the area is going to be the width of each rectangle. So each rectangle is a with pyro or two. So the upper some is going to be pirate too. And then you evaluate the first height of the first rectangle is gonna be f of pie or two and the same for the second to make it an upper some. So it's gonna be f pi over two plus f pie or two and it pi over two. This is an easy one. The function evaluates the 3333 plus three is six. So you've got six pi over two. So this just turns out to be three pi now to find the lower some for that same case to find the lower some. Again this thing starts out increasing. So what would happen if I switch ah to left sums? So to find that lower some, you can see like okay at the height of that first rectangle has to be the left end point and the second rectangle has two instead of being the left hand point, it has to be the right in point. So it's going to be F zero and F of pie to get a lower some. So the lower some is going to be same wit. Okay And then you're gonna have F0 plus F. A pie and if you look at f of zero is two and f of pie is to say two plus two is four. So this is four over to this is two pi. So a big difference between the two here and in this case now let's just go and take a look and say what would happen if I increase the number of rectangles, you should increase uh your accuracy by doing that. So let's go to end equal four. So if N is equal to four and then let's just go back to doing the upper some first. So again you can see that I will get an upper some between zero and pi over two by using the right in point. But in order to get an upper some on the rest of that power over 22 pi. I must use a left endpoint. So that's going to say I'm going to use pi over four pi over two for the first two rectangles. And then pi over two and 3/4 pie for the second to So what I see here, this is going to be the width of the rectangles. Is still if you look at this one because it's of number is four. If I look at that because of his for each one of these is pi over four. So it's going to be pira four. Mhm. And then you're going to have f of to get an upper some you're using the right sides of pi over four empire or two. So pi over four plus f higher or two. And then on the second two rectangles you're going to need to use instead of the right in point. The left endpoint power over two and 3/4 pie. Yeah. Mhm. Mhm. And that will give me the upper some For the case of in equal four. Uh huh. Now in the case of the lower some for that same scenario. So instead of writing points let's switch and look at what the left in point to do. And you can see indeed the left endpoint will give me a lower some for the first half but it gives me an upper some so the left endpoint will be used on the first two rectangles and then I need to use the right in point on the second to so I'm going to use zero empire A four in the first case. So it's going to be pira for zero plus. So it's gonna be zero empire over four. And then for the second to in order to get an upper excuse me a lower some I'm gonna be using the right so it's 3/4 pie and pie. Yeah. Yeah. Mhm. Yeah. Oh. Mhm. Yeah. And so that will give me the lower some in that particular case. Now if we look at what happens when in is equal to eight so I'm going to go to more rectangles. It's still the same scenario. You see that in order to get an upper some. So in order to get an upper some I have to start out on the first half of this curve using right in points. And then I switched to using left in points. So at pi over 854 3/8 and pi over two that's we're gonna be using the right in points. So if I go back here and now if I look at it, pi over eight is the width of each rectangle. So this is going to be high over eight. Mhm. You're gonna have f of however eight plus F Of go back and look to its pyro a pyro force you're counting by eighths so pie. Before. Yeah So to age plus 3/8 And then when you get to the last one at pirate to uh so I'm using I'm using the right in points, the pirate party for 3/8. And then yes five or 2. And then continuing that now for the second half I'm still gonna be counting by eights. But instead of using right in points, I'm gonna be using left in points to get an upper some. So started pi over two. So F However to which is 4/8. f of 5/8 pie. Yeah. Plus f. of 6 8 pi Okay. Mhm. And then it ends with 7/8 pie. Mhm. Mhm. So that's how you'll get an upper some. Using eight rectangles. Now to get a lower some in order to get a lower some go back and look at our graph. Uh to get a lower some we're going to switch. So if I were using right in points can be left in points. I can see that's a lower some for the first half. But I would need to switch from using those left in points to using right in points for the second half. So I'm still counting about higher or eight, but I'm starting at zero. So what that means is the lower some is going to be however eight. And then you start at zero F zero. You count by pi over eighths. So 1/8 Plus 2/8 pie plus 3/8 pie. They go look at our graph. So I got to 3/8. And now when I get to the next one of pi over two instead of using the left hand point, I'm gonna need to use the right. So I need to start at 5/8 and count by eights. So that's gonna be f Of 5/8 pie Plus. f of 6/8 pie Plus F of 7/8 pie. Yeah Plus F of 8/8 pie. Which is F of pie. So that's how I will get the same. Get the lower some for eight. Okay, now the you can use a calculator to input this uh to get your estimates in a decimal form, but this is basically how you would do the upper some For the case of in equal to in equal four and in equal eight.

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