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Question

Use the Midpoint Rule with the given value of $ n $ to approximate the integral. Round the answer to four decimal places.

$ \displaystyle \int^2_0 \frac{x}{x + 1}\, dx $,     $ n = 5 $

Frank Lin · Accepted Answer

so use the midpoint rule and approximately rock tangos in the proper night. Off this section, I explained the general formula, and you can use their former and plug in lower Barbara, Pa and and B, and this&#x27;s your F and general value. And you put those into the general formula and you can get an estimate. And at the end for this part you have to use your calculator, which on TV to you.

Robert Daugherty · Answer

Yeah, yeah. So five dot to problem number one. So we&#x27;re asked to use Riemann sums to estimate this? Uh integral. So our estimate is going to be using five rectangles. So in the interval From 0 to 1, If you divide that into five, everyone is going to be .2. So point to 0.4 0.6 .8. And then one. Now we&#x27;re also asked to use mid points. So what are the midpoint is going to be? The mid points are going to be .1 0.3 0.5 0.7 and 0.9. So if I were to estimate this with rectangles, I&#x27;m going to use the function value at the midpoint. The width of each rectangle is still going to be .2. So my estimate Ah for this integral. So the integral from 0 to 1 Cube root, x cubed plus one dx is going to be estimated by the area, which is the sum of these five rectangles. So what is the width of each rectangle? So the width of each rectangle Is going to be what one? So I got the width is one divided by five. So the width of each rectangle is 1/5. Yeah And then the height of each rectangle is going to be evaluating at the midpoint. So you&#x27;re going to get .1 Q plus one. And let me just do it sort of this way plus the square root of 0.3 cubed plus one Plus the square root of .5 cubed Plus one plus the square root of .7 cubed plus one Plus the square root of .9 cubed plus one. So all of that turns out to be our estimate. Now there are easier ways of doing this with a calculator than doing all of that out. Let&#x27;s try that. So let&#x27;s just create my function is the square root. So the square root no. Of X cubed plus one. And I&#x27;m going to store that my function F of X. And so now what I need to do is just create the sequence of all of those mid points. Or I could just on it just sum them all up. I could just say, hey, the area is going to be, What is it? 1/5, so 1 5th Times F of .1. Yeah. Plus F point too. Excuse me, F A 0.1. And then fo 0.3 Plus F 5.5. Yeah, half of mhm. Seven F 0.9. So 1/5 Uh f of one F 2 F three f of five. F seven F. Of nine. So 12345 is the case there. And so what you see there is 1.109 6 7. So this area estimate Is about 1.109. Let me make sure right, that right? 1.109 67 Okay, so that is the estimate of this integral.

Amrita Bhasin · Answer

this question asked us to use the midpoint rule to calculate um before the first step is to calculate Delta X using the formula B minus. Stay over at playing it. We end up with 0.2, which means we can now plug this end to our formula. Exit by is a plus. I Delta X 0.2 times half of 0.1 plus half of 0.3 plus half of 0.5 plus F 0.7 plus half of 0.9. This gives us 0.3084

Fuzail Shakir · Answer

Okay, so we have a is equal to zero. B is equal to five and N is equal to four. Delta X is equal to five, divided by four, which is 1.25 So using the formula of X sub, I&#x27;s equal to eight times I Delta X. We have the mid points off 0.0 point 0.0 6 to 5, 1.87553 point 1.87553 1 to 5 and 4.375 So therefore, using the midpoint rule, we take F off each of these and then multiply it by 1.5. So we end up with 1.25 multiplied by zero point 2091 plus 0.5391 plus 0.4291 plus 0.2409 which is equal to 1.25 times 1.4182 which is equal to 1.7753

Bobby Barnes · Answer

so everyone used the midpoint formula to approximate this. Um, what I first do is just figure out what Dell Texas. So over here we have the formula for that. So r B is pi r a zero and that our in is going to be four. Or at least let Chuck Yeah, in this for So there&#x27;s just gonna give us pie fourth. So now there&#x27;s two ways to do. Says we&#x27;re doing the midpoint formula. Um, we could just kind of use like the equation that I have a top here or what I normally like to do is first create like the set of partitions, like we would be doing the left or right. Riemann sums and then just average them. So I&#x27;m going to do that. So it&#x27;s going to start from zero and then remember to get the next X I. We just had Delta X two. It would be pie, fourth pie, half three, pie fourth and then pie. And so now, So this is just a set of partitioning, and then the ones for the mid point we just average thes, so the average of that would be pie eight to the next one will be three pie eight five Pi eight and then seven Pi aids. All right. And so now if we want So since Delta X is just gonna be pi for you just kind of pull that out front of this So it would be pie fourth times that we would just plug in pi a three pilots and so on. So high eights times sine squared of high plus three I ate signed square of three pie eight and then plus five pi eights plus plus sine squared of five high eighth and then plus seven pi eight sine squared of seven pi tight. Right Then we just need to plug this song into a calculator and then get our estimate. So let me see what we get. So it&#x27;s like approximately so pi fourth times, uh, 3.141 592 And then just gotta keep on going that we would need to multiply this by pie and then divide by four. And then this would be approximately 2.467 Or and I believe they wanted it to go to four decimal places. So this will be approximately integral from zero to pi of x times sine square of X x

Bobby Barnes · Answer

So if we want Thio, use the midpoint formula for this, Um, all we really need to do is first to find what is our Delta X? So remember, Delta X is supposed to be B minus a over and, um, so are being This case is a r a. Is zero and the number of victims we want US force that&#x27;s going to be, too. And then we need to find our set of partition. So we could either just use, like, this formula here, um, to get them. Because remember, each X I is going to be x I minus one plus Delta X So we could go about doing it that way, or we can just create our first set. I normally like to create the single like the set of partitioning points, and then I would just find the midpoint of all of those and I can just plug them indirectly. That&#x27;s the way I&#x27;m going to kind of do this. So, first, if we were department partition, this normally. So we start from zero a, and they&#x27;re just gonna keep adding Delta acts like we had here until we hit B. So it would be 0246 and eight. And then we have this new partitioning set, which is going to be for the midpoint. So then we just average these. So the average of zero and two is one. The average of two and four is three or in 656 and 87 And so these are going to be the numbers that we would end up plugging into here. So let&#x27;s go ahead and actually, um, set this up So our Delta X in this case, it&#x27;s just going to be, um too. So was kind of rape this out. So be to times and then our f is what we have on the inside here, So it&#x27;s going to be a sign of Route one plus sign of Route three plus sign of Route five plus sign of Route seven that we could just go ahead and plug these into a calculator on. Actually, heart went ahead and found these earlier, so this would be to and then 0.84147 plus 0.98703 and then plus 0.78675 and then, obviously for 70.47577 And then we just need to multiply all these together or add them all together than multiply. So that would give us eso approximately 6.18 e o. Actually, how many do we need to round 24 decimals? So that would be good enough. So this is what we would approximate. So this is approximately equal to the integral from zero a. A sign of the square root of X dx.

Frank Lin · Answer

to use them in point rule. The first thing to do is calculate Delta acts was using the formula. Do you mind? Stay over on this. Gives us 2/5 or one or 0.4. Is it just more so? Now we know we have affects is X over Expose one at the interval. Mid points. So the integral from zero to X over exports one D ox. We know this is gonna be 0.4 times up of x zero, but you can plug in to get 1.60 point 166 plus F of X one was F of X two pushed off of x three plus f of x four to give a solution of zero point 9071

Frank Lin · Answer

Yeah. So in the problem you see here, problem number 11, we have a definite integral and were asked to estimate this using uh rebound sums and rectangles, with the number of rectangles being five. So a crude approximation. So over the interval from 0 to 2, So two divided by five, his point four. So when I divide this up, you&#x27;re going to have 0.4. Yeah 0.8 1.2, 1.6. Now were also asked to use the midpoint rule. So for mid points, the mid points are going to be a point to 0.6. Um 1.0, 1.4 and 1.8. So this integral From 0 to 2 of X over X plus one dx can be approximated by The areas of these rectangles. So each rectangle is a width .4, so 0.4 times. And now we have to figure out so it&#x27;s going to be substituting each of these mid points into this formula. So it&#x27;s going to be 0.2 over um actually just gonna be 1.2. So let me just do it that way, one plus 10.2, so point to over 1.2 Plus .6 over 1.6 plus one over to Plus 1.4 Over 2.4 plus 1.8 Over 2.8. So that is going to be my area approximation, so I can just do all of that on my calculator. So it is .4 times. And so we had what .4 divided about 1.4 Plus .8, divided by 1.8 plus um one divided by two, so one half Plus 1.4 Over 2.4 plus 1.8, divided by 2.8 98254 is approximation, so 98254 is my approximation. Yeah. And we can see how accurate that is. We&#x27;ll find better ways of doing it ourselves. But I mean what if we were to ask for? Yeah. The integral From 0 to 2 of X over X plus one, So .90139. So I&#x27;m in the ballpark, but not a great approximation.

Problem

Use the Midpoint Rule with the given value of $ n…

Problem 11 Medium Difficulty

Use the Midpoint Rule with the given value of $ n $ to approximate the integral. Round the answer to four decimal places.

$ \displaystyle \int^2_0 \frac{x}{x + 1}\, dx $, $ n = 5 $

Answer

0.9071

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