a-approximate-the-value-of-each-of-the-given-integrals-by-use-of-simpsons-rule-using-the-given-val-8

Question

(a) approximate the value of each of the given integrals by use of Simpson's rule, using the given value of $n,$ and $(b)$ check by direct integration.
$$\int_{0}^{2} x \sqrt{x^{2}+1} d x, n=4$$

Devanshu Khadka · Accepted Answer

So hello everyone today. We&#x27;re going to solve the integration from 0 to 2 X. Into the whole route of expert plus one where N is equal to four. In the good question. We are told to integrate this first for party using the Simpson&#x27;s rule and find the integral. And for part B we need to use simple integration to check the answer for party. So let&#x27;s start with party. The former for symptoms through is given at the top is is the integral from A to B F X G X approx mr Dell dx over three into fx not plus four. Fx one was to F X two plus +463 So it alternates between four and two and Fx And so except for the first and the last term, everything else is um multiply before and to an alternate being four and two. So to find little X. The formula is B minus A. R. N. And the government question B. S two which is the upper limit And is the door handle a mentor to zero Or end given n. s. four physical is still half. So now that we have the X. As half we can find the extremes. So X. Note will be A which is zero as it starts with zero annex one. As Uh huh. As it increases behalf every time extra would be one extra would be 312 X four. We can write their property export would be to do which is be no we have the um X points. We need to find the via points by plugging in the X. Points into the main equation which is X into root of X squared plus one. So when you plug in the first F. X. Not you get as um zero entered route of zero plus one that gives you zero. Um Then I&#x27;m gonna skip a step and directly find four X. One instead of just fx for fx one instead of just fx one. So let me just write that will take I&#x27;m gonna find four into affects one that it&#x27;s given in the formula. You can see it&#x27;s for it affects one as a second term. So four and two X. one. When you put half into the main integral equation you get do exercise so off. Uh as I wrote fire Then f. f. x. two is four. Okay that&#x27;s right too because the third time that I went to into the earth of X. One and then the third time the next door. And that&#x27;s the answer is stupid too. Then FFX three it&#x27;s the third time as for into the death of three by two Which gives you the answer is 3-13. Sorry for the background as if you can hear that and finally affects for is just of two which you get the answer as to route five. So now you have all the values zero Route 5-2-3 13. Now you need to plug it in the equation. So the formula is Delta X over three. I&#x27;m gonna write that. So delta X over three. We have given the death tax as one or two. So one or two Into three gives you one or 6. one or 6 into zero plus book five. Keep in mind that we have already moved there for and to an alternate between four and 2 to find the our final terms which have to be plugged into the equation. So that&#x27;s what we&#x27;re doing right now. Last two or 2 Plus three or 13 plus to route five. Now that you get the answer as But you get the answer is 3.39-2. And that is your final answer for part A. And that&#x27;s what you get after applying Simpson through. Um Now for party we need to prove this by prove that this answer is correct by finding the integration definite integral. So it&#x27;s definitely went from 0-2. I&#x27;m just gonna write it down. It&#x27;ll pick X. O. X. Where this one. Now to integrate this you would use U substitution where you would be Ruutel X square. So on the right that town except this one so you will be. Uh huh. So to expect this one and uh you would get the um now that now that you&#x27;ve taken us out of X. Purpose one your balance you can change your upper bound and lower bound and you&#x27;ll get the lower bound as one an upper bound as your five So on and five of you square plus t you. And then you need to integrate this and you get the integration as you cube for three from 15? Yes sorry not passive. Let&#x27;s see you want to come here because it&#x27;s a definite integral and um when you plug this into the equation you get five route five you can use the calculator to find the answer for this which is five or 5 or three. Sorry this thing is not working with uh 55 43 -1 or three. Which when you find it in the decimal farm We get 3.39. So you get the same answer as what he had gotten using the Simpsons room but this is definitely more accurate than Simpson role. So that&#x27;s how you check from normal integration that your Simpsons will did work. And so I hope you guys understood that the main part of this question was finding the um, in technologies in some side room, uh, and knowing the formula for it. And I hope you guys understood that, and thanks for watching.

Victoria Dollar · Answer

Okay, So for this problem we have in equals for and we have the function f of X equals, um two X squared plus one. And we&#x27;re on the interval between zero and to. So what we want to do is we want to figure out we want to create a table of all of our values. So x and f of X and we want to figure out how maney what the increments we&#x27;re going to be. So we&#x27;re going to take to minus zero, which is our interval and divide by four. So we&#x27;re going to get one half, so I&#x27;m going to do zero 0.51 1.5 and two. So if I were to plug in zero into f of X, I would end up with one 0.5 as X would give me f of X equals 1.5 when x is one f of X is three one x is 1.5. Ffx is 5.5 And what exes to ffx is nine. So now that I have that, I can go ahead and do the trapezoidal role. So the trapezoidal rule for this function is going to be a t of four equals. Um and then we&#x27;re going to do one half over to And then what we&#x27;re going to do is we&#x27;re going to do our f of X is on each of them. The outside, too. We&#x27;ll just be that f of x value. So this one will be one, and then all the inside ones will be multiplied by two. And so I&#x27;m gonna do two times 1.5 plus 2.3 times three plus two times 5.5 plus nine. Remember that outside one does not get anything multiplied by it. So then, if I so point 5/2 is going to be 0.0.25 than if I were to add and multiply all of these together, I&#x27;m going to get 30. So that&#x27;s going to end up being 7.5 and then for part B. So this is going to be, um, Simpson&#x27;s rule. So we&#x27;re gonna have s of four equals, and it&#x27;s gonna be point 5/3. And this one is similar to trapezoidal rule in the sense that it&#x27;s going to be the outside to We&#x27;re just gonna be ffx, and then the inside values. They&#x27;re gonna alternate being multiplied by four and two. So let&#x27;s start off with four and multiply it by 1.5 plus two times three and then go back to four times 5.5 and then the outside won again will not have anything multiplied by it. So I&#x27;m gonna have point 5/3. If I were to add and multiply everything together, I&#x27;m going to get 44. And so then this is going to be 7.333 repeating. And then last but not least, I want to take the integral from 2 to 0 of two x squared plus one DX. So I&#x27;m going to end up adding one to the exponents. And then don&#x27;t forget, I&#x27;m gonna multiply by the reciprocal of my new exponents. So then I&#x27;m gonna have plus one x from 2 to 0. So now I&#x27;m gonna have two thirds X to the third plus x from 2 to 0. So I&#x27;m going to go ahead and do my squared or do my two first. So, to to the third is eight. Um, I have 16 3rd plus two, and then I&#x27;m gonna have this will all be zero because I have anything time you have X multiplied by anything, it&#x27;s going to be zero. So then 16 3rd plus two is going to end up giving me. I believe it&#x27;s 22 3rd, which is the same thing as 7.3333

Victoria Dollar · Answer

Okay, So for this problem were given that in equals four and that f of x equals three x squared plus two and were asked to find out the area under the curve estimation wise, um, using the trapezoidal role in Simpson&#x27;s rule. And then we&#x27;re supposed to go back and find the exact value by integrating it. So And we&#x27;re also given that this is between that X is between two and zero. So what I want to do first is I want to break in down, So I want to go between, um so I kind of want to break this down. So I&#x27;m gonna do X and f of X, and I want to break this down between zero and two. So I want four steps. So if I look at this and due to minus 0/4, that&#x27;s going to give me 2/4 or it&#x27;s gonna be one half, So I want to do this in in intervals of one half. And so if I were to go back and plug this in s 05 X equals zero and plug this into ffx, I&#x27;m going to get to If I plug in one half into X. I&#x27;m going to get 2.75 for ffx. Um, f of one is going to be five f of 1.5 is going to be 8.75 and then f of two is 14. Okay, so now what I wanna do is I want to do the trapezoidal rule. So what I need to know is the formula. So it&#x27;s going to be teeth in equals the change in X over to of F of X of zero plus two f of X one plus two f of X two and then so on and so forth. And then plus two f of um x of in minus one. So we have to all in the middle, and then the two end values are going to be just without a two multiplied in front of them. So since in equals four for this one, we&#x27;re gonna have t four equals. Our change in X, which is 0.5 over to my f of X zero we said was, um was too. F of X one is 2.75 f of X two is going to be five f of X three is going to be 8.75 And then, of course, my last one is 14 which I&#x27;m not going to multiply by two. So then, after I So I know that point 5/2 is 0.0.25 And then if I were to multiply and addle of these together, I&#x27;m going to get 49. And so this is going to give me 12.25 Okay, so that&#x27;s the trapezoidal rule. And then we&#x27;re gonna also do this by Simpson&#x27;s rule. And so Simpson&#x27;s rule is kind of similar to trap his idol. So it&#x27;s gonna be, uh, changing X over three and then f of x of zero plus four f of x one plus two f of x two plus four f of x one So on and so forth. And then, of course, my last values it&#x27;s gonna alternate between four and two in between the first and last and then the first and last is just gonna be time without or multiplied by one. So there&#x27;s not committing multiplied in front of it. So for mine, it&#x27;s gonna be s four equals point 5/3, so f of X zero is, too, that I have f of X one, which is 2.75 And then f of X two is going to be, um, five. And then I&#x27;m gonna alternate with the four against, and then my next one is going to be 8.75 and then my last one is just gonna be 14 without anything multiplied to it. So then I&#x27;m going to keep this point 5/3 just because of the fact that it&#x27;s not gonna be a rounded decimal. If I add all of this together, I get 72. And then my final answer I get is going to be 12. Exactly. So now we want to integrate. So when I do the integral from 2 to 0 of three x squared plus two d x. Okay, so if you recall, I&#x27;m going to take my X squared, and I&#x27;m gonna add one to the exponents so it&#x27;s not gonna be a three, and then I&#x27;m going to take the three and divide by three because remember, actually, we do it this way, so I&#x27;m gonna multiply this by one third. Remember, I wanna multiply by the reciprocal of the exponents. And then, of course, to is simply gonna become two X. This is gonna be between zero and two. So if you notice my three and one thirds, we&#x27;re going to cancel some have X to the third plus two x from 2 to 0, and I&#x27;m going to do the two&#x27;s first. It&#x27;s gonna be two to the third, plus two times two minus the zeros. So zero to the third, plus two times zero. So if we notice to to, the third is going to be eight plus four and then minus zero, so that just cancels. So this is going to be 12.

Victoria Dollar · Answer

Okay, So for this problem were given that in equals for our ffx is X times the square root of two x squared plus one. And this is over the interval between zero and four. So the first thing I want is I want to kind of collect a few pieces. Then we&#x27;ll talk about break down each approximation. Of course, the integral later kind of as we go. So I&#x27;m gonna have X and f of X. So I want to find out what? How I&#x27;m going to get from 0 to 4. So, like, what Increments? Some gonna take him a subtract that the interval to find out how big the interval is and then divide it by in. So four minus zero is for over four, which is one. So what do these from in increments of one. So if I plug the&#x27;s in, I&#x27;m going to get zero square root of three six, three square root of 19 and four square root of 33. Okay. And then for the first one, we&#x27;re going to use trapezoidal rule, so just kind of as a refresher. This is our approximation. So we&#x27;re going to use Delta X which is the one that we found earlier, divided by two and then our first and last f of X, um, are going thio, not have anything multiply to them and all of the others will have to multiply to them. Okay, so now we have to do is plug and we have our values. We&#x27;re just gonna plug them in so it&#x27;s gonna be 1/2 and then zero plus two square root of three plus two times six plus two times three square root of 19 plus four square root of 33. So this is gonna be 0.5 times, so everything in the brackets will be 64.5958 So this gives us a total of 32.2979 This would be the approximation of that curve in that interval. And then for Simpson&#x27;s rule, it&#x27;s going to be Delta X over three and again. Our first and our last are not gonna have anything multiplied in front of them. And the ones in between are gonna alternate being multiplied between four and to. Okay, so now we have our formula. Delta X is still going to be one. So then zero plus four square root of three plus two times six plus four times three square root of 19 plus four square root of 33. So this is still going to be one third. And then if we were to multiply everything and at everything together we would get 94.2132 so that times a third or divided by three, will be 31 0.4044 So that would be that approximation of the area under the curve of that interval. And then last but not least, we want to actually find the integrated amount, which is going to be the which is going to be the actual value. So it&#x27;s gonna be two x squared plus one d x. So this one, I used a use substitution, and I made it equal what is underneath the radical. So then the derivative of that would be four x so my I leave the values there and then I put it back later, so I&#x27;m gonna have X over four X. So the X&#x27;s cancel, So I&#x27;m gonna have 1/4 and now I&#x27;m going to do you to the one half, do you? So I&#x27;m gonna have 1/4. And then, of course, I&#x27;m going to add one to this. It&#x27;s not gonna be you to the three halves times the reciprocal of the new exponents. So two thirds times 1/4 is now going to be 1/6. And then, of course, my you is two x squared plus one to the three halves power. So if I were to plug in four, um, this is going to give me 16 33 to the three halfs power minus 1/6. Because if I plug in zero, that&#x27;s going to give me 1/6. So if I use a calculator and find out the value of this, I get 31.4284

Victoria Dollar · Answer

Okay, So for this problem were given that n equals four our f of x equals two X to the third plus one, and this is gonna be on the interval from negative 1 to 2. So the first thing I want to go ahead and Dio is set up my table so I can determine my f of X values. And then I also need to figure out my increments in between these intervals. So what I&#x27;m gonna do is I&#x27;m gonna go ahead and take two and subtract negative one and divide by in, which is for so this is going to be 3/4 or 0.75 So I&#x27;m gonna go from negative one up 3/4 to negative 0.25 and then up 3/4 again, 2.5 and then up again and then up till we get to two. So then, if I plug in the X values, I&#x27;m going to get as follows. So 0.96 um, nine. 1.25 4.906 and 17. It looks like a huge jump, but it&#x27;s actually it. It kind of jumps a lot around a lot. So the first thing I want to do is I want to use the trapezoidal rule, so it&#x27;s going to be the change in X over to times f of x of zero plus two f of x one plus two f of x two plus two f of x three plus f of x four. So now I have to do is plug values in. So we already said the change on X is that 0.75 that we found earlier and then I have negative one plus two times 20.969 plus two times 1.25 plus two times 4.906 plus 17. So then, if I add everything inside the brackets and multiply everything based on the order of operations, I get that it&#x27;s 30.25 So then my final answers. 11 points. 3438 So now to use Simpson&#x27;s rule, it&#x27;s very, very similar. So it&#x27;s going to be S four equals the change in X over three. So then my do ffx not plus four f of x one plus two f of x two, plus four f of x three plus f of X four. So now I have to do is plug values in. So the change in X is going to be that 0.75 value we calculated over three. So then I&#x27;m going to have a negative one plus four times 40.969 plus two times 1.25 plus four times 4.906 plus 17. So then, um, 0.75 divided by three is just gonna be 30.25 And then if I multiply and add everything inside the brackets, um, I get 42. So then my final answer is 10.5. So then, if I were to actually go in and sulfur the integral so we get the integral from two to negative one or negative 1 to 2 of two x to the third plus one d X So then I can I&#x27;m gonna have X to the fourth with my two in front and then remember, I want to Duthie new exponents. Um inverse. Multiply to that and then plus x from two to negative one two times 1/4 is going to be one half x to the fourth plus x from two to negative one. So if I look at two so to to the fourth of 16. Half of 16 is eight plus two minus negative. Once the fourth will be one half eso. Yeah, negative. One to the fourth. Power will be positive. One times. One half is one half and then minus one. So this is gonna be 10 minus negative, one half, So we&#x27;re gonna have a double positive, so it&#x27;s gonna end up being 10.5.

Victoria Dollar · Answer

Okay, So for this problem were given that in equals for f of X equals four x e to the negative X squared over the interval zero and one. So the first thing I want to do is I want to figure out what the X values we&#x27;re gonna be. Um, so that way I can go ahead and plug those into find ffx. So I&#x27;m gonna take one minus zero and then divide by in, which is four. So this is going to give me 1/4 so I&#x27;m gonna take zero 0.25 0.5 0.75 and one. So then, if I plug these values into ffx, I end up getting, um, zero 0.9394 and you will have to use a calculator on this. Ah, 1.5576 1.7094 and then 1.47152 Okay, So what, um, I want to do now is go ahead, jump into, um, trapezoidal rule. So for tea base, for we&#x27;re going to be looking at the change of X over to and then thief first f of X plus two times the middle three, Um f of x is and then we want to add the last one without anything multiplied in addition to it. So now we&#x27;re gonna do. Is this change of X? Is this 1/4 that I found? So 0.25 divided by two and then zero plus two times 20.93 94 plus two times 1.5576 plus two times 1.7094 and then plus 1.47152 So then, if I were to plug in everything inside the brackets, but you&#x27;re multiplying and adding everything I get 9.884 and then, of course, 0.25 divided by two is 20.1 to 5. So this final approximation is going to give us 1.2355 And then if we look at B so this is going to be Simpson&#x27;s rule, so s a four. It&#x27;s gonna be the same change in X we used, which is the 40.25 This time we&#x27;re going to divide it by three and again, we&#x27;re going to take our first F of X and then the middle three, we&#x27;re gonna alternate between four and two, multiplying them and then plus four f of x and then plus the last one which will not have anything multiplied in front of it. So again, we&#x27;re gonna have this 0.25 over three zero plus four times 40.9394 plus two times 1.5576 plus four times 1.7094 plus 1.47152 And of course, if I multiply and add everything inside the brackets, I get 15.1819 which is going to be 1.265 So we have our two approximations, and now we want to actually find the integral of this. So between one and zero, we have four x e to the negative X squared DX, and so I&#x27;m actually gonna have to do a U substitution. So I&#x27;m gonna say negative X squared. So which means d you would be negative two x d x. So which means that I&#x27;m gonna have to divide this so I do for X over negative two x e to the u do you So which means this cancels and this will become too. So I&#x27;m gonna have from 1 to 0 negative to eat of the you do you. And so now I&#x27;m gonna have, um, negative to eat of the you from 1 to 0. So then I have negative to eat to the negative X squared from 1 to 0. So if I plug this in, is one I&#x27;m gonna have negative to over e minus. And if I put this to the zero is going to be negative, too. So it&#x27;s gonna be two minus two over E, which is 1.2642

Problem

Approximate the value of each of the given integr…

Problem 6 Easy Difficulty

(a) approximate the value of each of the given integrals by use of Simpson's rule, using the given value of $n,$ and $(b)$ check by direct integration.
$$\int_{0}^{2} x \sqrt{x^{2}+1} d x, n=4$$

Answer

Related Courses

Basic Technical Mathematics with Calculus

Related Topics

Discussion

Top Calculus 1 / AB Educators

Catherine R.

Samuel H.

Michael J.

Joseph L.

Calculus 1 / AB Courses

Precalculus Review - Intro

Functions on the Real Line - Overview

Recommended Videos

Watch More Solved Questions in Chapter 25

Video Transcript

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

Catherine R.

Samuel H.

Michael J.

Joseph L.

Precalculus Review - Intro

Functions on the Real Line - Overview

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

Basic Technical Mathematics with Calculus

Catherine R.

Samuel H.

Michael J.

Joseph L.

Precalculus Review - Intro

Functions on the Real Line - Overview

Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

Catherine R.

Samuel H.

Michael J.

Joseph L.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus