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Use the Midpoint Rule to evaluate the integral f z? In(r2 + l)dr41n(10) 2 1n(4) 2 In(2) 41n(5) 21n(10)...

Question

Use the Midpoint Rule to evaluate the integral f z? In(r2 + l)dr41n(10) 2 1n(4) 2 In(2) 41n(5) 21n(10)

Use the Midpoint Rule to evaluate the integral f z? In(r2 + l)dr 41n(10) 2 1n(4) 2 In(2) 41n(5) 21n(10)



Answers

Use the Midpoint Rule with $n=4$ to approximate the area of the region. Compare your result with the exact area obtained with a definite integral. $$ f(x)=1-x^{2}, \quad[-1,1] $$

We're going to calculate the meat boy rule approximation to the into girl from 2 to 10 off the function two times X square for and it was one that is once of interval for any girl stood to serve intervals and an equal four serve intervals. So we're going to start with once of interval. That means that we're gonna take, in this case, the whole interval off integration to 10. So, uh, we have eggs, not equals two x one equals 10. And if you want, we can say they will die X equal to 10 minus two over one. That is eight. That's the length of the interval. If integration, which is the oldest of in the road, you can see there in this case, that's why we only have notes. It's nautical stewed and X one equals stand which goes wants to be in points of the interval off integration. And so we know that the meatball rule in this case we're gonna call it in one is if at the meat point off the serve, inter off the interval for 2 to 10 times the lands of these air into all that is still takes because eight. So we re going to calculate the midpoint off thes of interval from two detent that is to bless. Stand over to that is 12 over to that. It's six midpoint off the interval to 10 and then and one is equal to if of six times eight in that ease. Now we get to name of use letter F already, but we have a written down. So we define f of X as integral to a X square. So here we're talking about 26 square times eight. And this means that the mid one rule for only one interval is and 576. So this is in one. We're now we're going to calculate for animals, too, to serve Interval. So Delta X, in this case, is Dan Manus to over to that. He's aid over to that is for And so we have this. The nodes define us. A, that is two plus Delta eggs is four times k for K running through zero. And what? Um, so you want into Sorry, I forget him. What? My brother Here It's just until end. That is an equal studio. Teoh one and two And with that we get, it's not equals. Two x one equals two plus still takes his six and eggs two is equal to six plus still Texas stand. So we have these three notes and we have to calculate the meat points off each of thesis of Interval certified. But the three notes that he's into a from 2 to 606 to 10. Me points are so, uh, first made boy nous tube six over to that. He's a door. It is four, and M two is equal to six. Plus, Stan over two is eight. So, uh, this are immune meet points we're going to use in the formula and two is then equal the midbrain rule. Fortress of Interval sees effort The first midpoint and one times that's eight. We're gonna put it outside because his common factor plus if at the second midpoint and to time still to eggs that these times for so we know that this is two times sorry. We're going to ride the values to tonight Currently. So this f of four plus if wolf, uh, eight times four. That is two times for square plus two times eight square all that time for So we have here, uh, four square 16 tends to 32 waas aids. Where is 64 times to ease 128 times four and that he's 160 times for and that it's finally 640. So we get that meet point rule using to serve intervals is 642 approximately given into girl. And now we're going to use four sub intervals n equals four. This case still toe Xs equal to 10 minus 2/4, eight and four. He school, too, and the notes picks a k are equal to two plus two times K that he's four k equals 012 three and four. So the notes bar, uh, to Plus it'll taxes four Pacific Tech. It's six plus in taxes h then 10. We have five notes this a year, and this sub intervals are 24 for six, six AIDS and 8 10 and we get a calculated meet points of all these force of intervals. You re put it here means points then. MM. For the interval to four is I am one is two plus four straight cars. The year. Sorry. Okay, he's am one equal. Two plus four over two is three and we have him to is war plus 6/2 is five in three years, six plus eight over to is seven and and four is eight plus 10 over to It's not so, uh, the meat point rule. Using four serve intervals is equal to if off m one plus f m two plus if aim three plus f of m four time still days, which is to that is the sequel to if, uh, three plus F five if a seven plus F nine times two and that is two times three square plus two five square plus seven. Sorry here. Two times I meant to times seven square last two temps. Nine square times to use a calculator is equal to 666. So in four, the midpoint rule approximation to the Given. Integral is in force of intervals sequel to 666. So we have t three approximations using the meat born rule. So the general idea is to calculate the notes. First we calculate the Dell Tex for that abused the whole interval length and the numbers of intervals. Then for that with that use the expression x of cables, a bliss que tenancy of X and without because the notes that defines the seven Toral sand for each one of them recalculate the mid points. And after that we know that the rule is a Some of the images of all those mid points multiply by Delta X.

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's just gonna give us pie fourth. So now there's two ways to do. Says we'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'm going to do that. So it'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'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

All right, So for this question, we have f of axe is equal to the internal from 0 to 2 off two acts e to the power of negative X t X. So if we find the width of the internal sensors of recalculate Delta X remember the formula that is B minus eight by Anna. So this is equal to two mighty zero and N is equal to 12. So our interval with is approximately 1.60 point 16 Right. So now we have to generate a table off effects at interval boundaries. So we're gonna just generate a table here, uh, at the intervals of 0.16 So x zero we have. If X equals zero at 0.16 we have 0.28 and 0.33 We had 0.47 It's your 0.5. We have 0.6 016 We have 0.68 on and 0.83 We have 0.72 one. He has your 0.73 there. We're gonna continue to keep doing this, uncle you of and is equal to 1234567812456789 10 11 12 13 14. All right, so just just keeps going on. Can you have four TT turns? We're gonna write out the trap is a little rule for 13 turns and then fill in the numbers. We could do that on the next page. So the drop is a little rule is approximately equal to Delta X, divided by two finds up of x zero plus two times f of X one plus two times aftereffects too all the way until two times f of x 11 plus f of x 12. So using the table that you didn't wait, we did in the previous page. You're gonna use those to Philip the numbers here, and they're gonna get NASA approximately equal to 1.18274 Next, we're gonna write out Simpsons Rule for 13 items on. We're gonna fill that out as well, using the numbers from the table. Remember that Simpson's rule is equal to Delta X, divided by three of athletic zero plus four f of x one plus two x two plus for FX three off x three and so on and so forth until you get to you four off x 11 plus f of X 12. All right, so if you saw this out, you're gonna get an approximate answer off. 1.18796 All right, So for the mid point, we have to translate half of X at the midpoint of each of the intervals. Onda again, if you that is calculating Delta, X and FX at now. Intervals of 0.83 become 0.3251 support until you have n is equal to 40 minutes. So for the midpoint rule, um, you're then gonna plug it numbers into Delta X times after of X zero plus f off X one all the way until plus half off X 11. All right. And you're gonna get an approximate answer of 1.19061 Right, So that right there is your final answer. If you want to use a graft from utility to confirm, you can use a graphing utility and the graph your graphing calculator will say that the eternal from zero to off to X e to the power of negative, X X is approximately equal to 1.1879 nine. All right, puzzle

And problem. 17. We're finding the value of this function f of tea on 0 to 2, using four some intervals of equal length and evaluating f it the sub interval mid points Joan zero to we know that our our intervals will be broken up by zero 0.511 point 52 So our mid points that we want to evaluate our function it our 0.25 point 751.25 and 1.75 And with 47 arrows on zero to our Delta X or hear Delta T is going to be 0.5. And if you want a little more explanation on average value, go back and watch problem number 15 and explains it clearly. But what we're gonna end up here is an area is going to be, um, Delta X or hear Delta T, which is your 0.5 gets to minus zero, divided by four times. They're for 0.25 f of 0.75 f of 1.25 and F 1.75 And so, if you go through unplugging 0.25 for tea, get out of a for F 1.75 for taking out of a for half and do that for all four of these. A pause the video here and do that, and we'll come back right back. Well, interesting. An interest rate. Use me. Interestingly enough, each one of these values is one because, um, end up with sun squared of power for which is one when you don't have a 1 to 5 three power for still one. No sign squares. Still one. So sorry about that positivity of It's pretty simple. It's just one plus one plus one plus one. So 0.5 times for equal to If this is our area under the curve, we have a the length of our sub interval or excuse me. The interval that we're concerned about is two units long. We have area of two with this side is too. So our average value of the function is going to be, um, one. And that makes sense because each one of these that we've evaluated for every one every value that we plugged in for tea. We get one each time. Well, obviously our average value for that function using those numbers is going to be one and later said, If you need more explanation on how what the average value is and what it means physically and how you look at it on the graph, go back and watch problem number 15 and I explain it much more in depth and more clearly.


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