if-your-cas-can-draw-rectangles-associated-with-riemann-sums-use-it-to-draw-rectangles-associated-36

Question

If your CAS can draw rectangles associated with Riemann sums, use it to draw rectangles associated with Riemann sums that converge to the integrals in Exercises. Use $n=4,10,20,$ and 50 subintervals of equal length in each case.
$\int_{1}^{2} \frac{1}{x} d x \quad$ (The integral's value is about 0.693 )

Regina Hays · Accepted Answer

okay. What we want to do is we want to show using a graphing utility types that the inner Groll from 1 to 2 of one over x t x um approaches 0.693 And we want to do it by doing the Riemann sums for in equal to 4 10 2050 And so what that means is, as we increase the number of rectangles that we divide the area into that that value the values will approach this 0.69 three. And so we&#x27;re gonna switch to a graphing utility. I have, um, Dismas, which is a free online tool that you can sign up for, and I&#x27;m going to change it toe one divided by experience. My function and we want to go from the left in point was a positive one. The right in point was a positive coup. So we&#x27;re only interested in this region right here, and the number of intervals is the number of rectangles or partitions. So we want to start with four partitions are now. I have it set up where I am actually doing the right hand. Um, some Cassie&#x27;s one, which is going to for a decreasing function. The right hand. Some is actually gonna be my lower son. I can actually change this to do the left hand some which is going to give me my upper some. And so let&#x27;s go ahead and do that. And so I notice for four rectangles or four partitions. My upper some is actually point um 0.759 five and we want to see is if I increase the number rectangles. So my change this to 10 that does my value. My integral approximation. Does it approach 0.693 and so it is getting smaller. So I went from 0.795 2.718 Now let&#x27;s see if we change it to 20 we get 200.7058 So it is decreasing on a let&#x27;s do 50 right angles. So sure enough, it went around 2.6 98 And so it is converging as I increase the number of rectangles as I&#x27;m increasing the number of rectangles and a mark partitioning my upper. Some is approaching 980.693

Gregory Higby · Answer

Okay. So again, you can pull up this program. Um, you know, illustrating remind sums. You can get a nice visual here. Just go ahead and type in ffx as one of her ex change left. Bound to be one. You&#x27;re right about to be to these air left. Remind Psalms, by the way, cause I&#x27;m using the let&#x27;s decided, uh, rectangles heights for the rectangles. You can see I have four rectangles, and as they add together the area of the four rectangles reappoint 760 because they do ask for the approximation. Um, but you can actually visualize as I increase my counter to 10 rectangles again left remind sums. Here I get an area of about 2.719 as I add all these rectangles together. And you can the more rectangles you get, the better the approximation. Because you can see I&#x27;m getting closer and closer to the actual area. 0.706 for 20. And then the last thing is to go up to n equals 40. Sorry. They want you to go toe en equals 50 and 0.698 is about as close as we&#x27;re gonna get now. I could go further. I can go to 100 if I wanted to. But what I like about this problem is that also gives you the exact area. Um, with this program by typing it in this way 0.693 which confirms what they said, the answer should be.

Regina Hays · Answer

Okay. What we want to do is we want to use a calculator or a program that will draw some agreement re Monson&#x27;s for us, and we want to show that the integral from zero toe one of one minus x t x converges to 1/2 and we want to do it for in equal four, 10 20 and 50 sub intervals. And so what we&#x27;re gonna do is I&#x27;m going to switch to, um, a graphing utility, um that is capable of drawing those rerun sums. And so it&#x27;s Dez mo&#x27;s. Um, if you have a t I 84 there is a special program that you can actually download. Um, that will also do this for you. And so we want to First of all, um, changed this to, um one minus X. Um, and so it&#x27;s a line man are left in point on the integral waas um zero and our upper limit waas one. And the number of intervals is for the first go round is a four. And so that&#x27;s what the drawing would look like if I had four rectangles. Um, and this is doing the left in point some. And so, um Yeah, and you could change it. This is a pretty interesting one, because you can change it to Dio. And this would be for a decreasing function. The left in point would be the upper some, and if I change it to the right in point, it would be the lower some in this case, we&#x27;re just gonna keep it said at the upper some, um and then also this allows me to also do the midpoint. And so we&#x27;re just gonna keep it at the left, handsome and for the left hand sound for decreasing function is going to be your upper limit or your upper sound. And then the integral that we get when we do the value of the integral when we do for four rectangles is 40.6 to 5. Um, which is not close to 1/2 but it&#x27;s getting there. And so what happens now if we just change the number of rectangles to 10 7 novels? And so now we&#x27;re doing 10. And so you notice now that the value the integral role goes to 0.55 Now, what happens if we dio 20? And so here we have 20 you know, girls and we get 200.5 to 5. So it is converging or it is approaching. The more as the partitions of the right tangles get smaller or the number of rectangles gets bigger, then that in a goal, is approaching 1.5. And so now let&#x27;s see what it does when we do 50 rectangles. Yes, point Dr One. And so it is approaching 1/2. So that&#x27;s what we&#x27;re doing for this problem. Dez knows. It&#x27;s, um, free. You just have to sign up for it. Or like I said, if you have a graphing calculator and inspire or 84 I know the 84 you can get a downloadable program to do the exact same thing.

Regina Hays · Answer

Okay. What we want to do is we want to show that the integral from negative 1 to 1 of the absolute value of X dx converges to the value of one. And we want to do it by using a graphing utility that does remind sums for us. And we want the number rectangles to go to 4 to 10 to 22 50. And hopefully, as we increase the number rectangles than the value of our remote and son approaches, um, one for us. So I&#x27;m a switch to a graphing utility I&#x27;m using Dez, Mose, um, and Dez Mo&#x27;s actually provides me away to graft the function. Um and so my go ahead and grab the function. Um, and it is going to be the absolute value of X. So there is my function of value of X, and it provides me a way to put in the left in point, which we had a negative one. The right in point of a positive one and the and also get let me get this out of the way. It provides is me away as a way to change the number of right tangles. And so we&#x27;re starting with four rake angles. And there we have it. And we&#x27;re doing I&#x27;m I&#x27;m doing the right hand some because I have, see, set to zero If you wanted to do, um I mean, see, set toe one. If I wanted to a left hand some, then I would have c set to zero. Or I can even do the midpoint some and let CB 00.5. So this program actually provides me the capability to do a bunch of stuff, and then the integral approximation is down here. So it shows me that for four rectangles, that value actually equals one already. And so let&#x27;s see if we change it to 10. And so I&#x27;m feel getting a one. And so this might be the case where the resolution of be utility might not just give me that ability. So in all three cases, or it could be that no matter what, how many rectangles I pick, I get a one anyways, so either way as it the number of rectangles increase, we did get a value of one for that integral

Regina Hays · Answer

Okay. What we wanna walk through is being able to show that the integral from zero to pi over four of seeking squared of X dx converges toe one. And we want to do it by using the Riemann sums. And we want the number of rectangles to be for the first time, then 10 then 20 then 50. And so we want to show that as we increase the number of rectangles, then the value that we get is approaches one. And we&#x27;re gonna do this by using the Riemann sums in a, um, graphing utility. And so I must switch over to the graphing utility and I&#x27;m using Dez knows which is a free online utility, and it provides me away to actually graph the function. Um, it actually does the number of rectangles, um, and then actually calculates that remind sums for me. And so a t I 84 there is a special program that if you have a t I before that, you can upload, and that will do the exact same thing. And I&#x27;m sure there is one for inspire a swell. But remember, however, has got this set up. So remember, um, you can&#x27;t, um, typically graph a Seacon square. So we have to change him into one of her co sign square and our left in point or lower limit of our integral was zero. So I may change that 20 are on upper waas a pie over four. So I&#x27;m gonna change that. And then here is I school down on the number of inner roles will be the number of rectangle. So we were the first thing we wanted to do was a four rectangles, and in this case, I&#x27;m doing the the right in point room on some which is going to give me the upper some. And you can change it by changing the value of the sea to either do the left hand, which, for this case, would give us the lower some or even the midpoint some. And so then I can scroll down. And here is the value of that re Monsanto for four rectangles. And so that is 1.11 one. And so that is close to one. So let&#x27;s see what happens if I change the number rectangles to 10. And sure enough, my value of my Rayvon son goes to 1.4 this time. Now, let&#x27;s see if we do 20 and now it decreases to 1.2 So my value of my reminds, um, is in converging to a value of one. And so now let&#x27;s do 50. And sure enough, I get 1.1 So you notice as I&#x27;m increasing the number of rectangles and you can actually see it in the picture on your on the screen as well that the number of rectangles increases the value the room on some or the integral actually converges toe one.

Regina Hays · Answer

Okay. What we want to do is to on a computer aided software is to be able to, um, use the Riemann sounds, um, to see if the integral ah x squared plus one d x converges to 4/3. And we want to do a number of rectangles going from 4 to 10 to 22. 50. And hopefully as the number of rectangles increase, the actual value that we get from the Riemann sums converges to 4/3. Um, so let&#x27;s go ahead and switch to a utility. Um, and I am using Dez Mose um, which is a graphing utility. You can do the same thing on on 80 i 84 if you have one or, um, or in inspire. I know for the 84 they do provide some program uploads that will provide you to do this. And so first thing we do is change it to what our function was. And our function was, um, X. It&#x27;s weird plus one. And so the earth or function, um, we&#x27;re actually going from the left in point is zero, which would be are lower limit, and our upper limit on the integral was one as well. So that would be all right in point we are actually going to be doing, um we&#x27;re going to start off with a number of intervals of four on. So there&#x27;s four rectangles. Now, this one is doing the left, um, in point for your rectangles, which is going to give you the lower some, Um, let&#x27;s go ahead and change it to where we get the upper some. So we&#x27;ll get kind of give us the upper limit. Their, um In order for me to do that, I need to change, see toe one, and you notice when I did that, the right tangles now are on the outside of the function and will give me the upper. Some on this will also do the midpoint. And so this tells me that thieve value the integral when I do for rectangles using the upper some or the right in point is, um one point 46875 and I were trying to get to 1.33 so we want to hopefully converge to 1.33 And so now let&#x27;s go up here and change our, um, number rectangles to 10 and it. Changes are integral to 1.385 And so you notice. Now we have the number of right tangles start increasing. So the width of the rectangles were getting smaller as a number of rectangles increase, and now we&#x27;re going to change it to 20. Man. Sure enough, we get 1.3, um six, which before we had 1.38 So it does look like it&#x27;s converging toe 1.33 or 4/3. And so now let&#x27;s change it to 50. And sure enough, we get 1.34 so it is converging to the value of 1.33

Problem

Use a CAS to perform the following steps: a. Plot…

Problem 94 Medium Difficulty

Answer

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Joel Hass, Christopher Heil, Przemyslaw Bogacki

University Calculus: Early Transcendentals

Anna Marie V.

Kayleah T.

Caleb E.

Kristen K.

Precalculus Review - Intro

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University Calculus: Early Transcendentals

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Joel Hass, Christopher Heil, Przemyslaw BogackiUniversity Calculus: Early Transcendentals

Anna Marie V.

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Caleb E.

Kristen K.

Joel Hass, Christopher Heil, Przemyslaw Bogacki

University Calculus: Early Transcendentals