use-the-midpoint-rule-with-the-given-value-of-n-to-approximate-the-integral-round-the-answer-to-fo-6

Question

Use the Midpoint Rule with the given value of n to
approximate the integral. Round the answer to four decimal
places.
$$\int_{0}^{1} \sin \left(x^{2}\right) d x, \quad n=5$$

Amrita Bhasin · Accepted 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

Robert Daugherty · 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.

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

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.

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

Fuzail Shakir · Answer

obviously using the midpoint rule to find fours and keeping in mind that ex eyes equal to ay plus Delta X I. You can end up with four, which is approximately equal. +045891

Problem

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

Problem 11 Easy Difficulty

Use the Midpoint Rule with the given value of n to
approximate the integral. Round the answer to four decimal
places.
$$\int_{0}^{1} \sin \left(x^{2}\right) d x, \quad n=5$$

Answer

$M_{4} \approx 0.3084$

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$M_{4} \approx 0.3084$

Biocalculus Calculus for the Life Sciences

Catherine R.

Anna Marie V.

Caleb E.

Michael J.

Integration Review - Intro

Definite vs Indefinite

James StewartBiocalculus Calculus for the Life Sciences

Catherine R.

Anna Marie V.

Caleb E.

Michael J.

James Stewart

Biocalculus Calculus for the Life Sciences