solve-each-given-problem-by-using-the-trapezoidal-rule-int_03-fracd-x2-x2ln-2-approximate-the-value-

Name: solve each given problem by using the trapezoidal rule int_03 fracd x2 x2ln 2 approximate the value
Uploaded: 2020-02-12
Duration: 30 s
Channel: Numerade
Description: Solve each given problem by using the trapezoidal rule. $\int_{0}^{3} \frac{d x}{2 x+2}=\ln 2 .$ Approximate the value of the integral with $n=6 .$ Compare with $\ln 2.$

Question

Solve each given problem by using the trapezoidal rule.
$\int_{0}^{3} \frac{d x}{2 x+2}=\ln 2 .$ Approximate the value of the integral with $n=6 .$ Compare with $\ln 2.$

Yuou Sun · Accepted Answer

problem. 14. This is just another punk days off calculation. There&#x27;s no tricks here. Just calculate average, change everything. It&#x27;s already know it. Here&#x27;s a function and their taxis three minus two, divided by 10 days. One or 10. Right. And here the T have a nasty just calculate, calculate and calculate. Everything is already knowing you could gather crackdowns. They&#x27;re here.

K B · Answer

So in this question, we&#x27;re gonna be using the trap is way cool. So are starting an agro he knows brings your own and spirit of four plus X cubed, Yes. Oh, and that is equal to six. So the way we&#x27;re gonna be finding our first piece of information is we&#x27;re gonna be doing Changing axe is gonna be equal to G minus a, which in this case is gonna be three b say, and then over and which is sex? So this is 0.5. Okay, so now we know that our the intern are questions basic, maybe 0.5. And this is Tee. We&#x27;re trying to find TN or in this case, it&#x27;s cheese. Six. That&#x27;s so it&#x27;s gonna be 0.5 over, too. And then this Times F zero and this is what we&#x27;re gonna do here is we&#x27;re gonna do cost, too. And the next thing we&#x27;re gonna plug into our function is gonna be zero cost 0.5 and we keep about actually doing this. We&#x27;re just keeping on adding 0.5. So the next one will be to out for one, and then we keep going to after 1.5 class to Africa, too. And then we would have plus two out of 2.5, and then we finally end up with plus after three. There&#x27;s no coefficient on the first and last from this room. So sorry, this little bit cramped, but I think I didn&#x27;t describe what we have to do here. So basically, we could just solve this out. We would plug these into our original function, and the answer that we end up with is 9.330 So this is gonna be our answer to this. 9.330

K B · Answer

so in this fashion were asked to use The trap is oil also. First, we&#x27;re gonna find this value, which you&#x27;re gonna be mind say, over. So we have one time for this Because we have, um this is equal in three over sex. It was three months here on display that says so. We have this values one hot. So now we know what they&#x27;re for, what we&#x27;re gonna have 1/4 times and that f zero plus. And now we look at our changing changing value, which is 1/2. So every time where I wanna wrap so it&#x27;s gonna be too of 0.5 and it&#x27;s gonna keep going. All the anto to half of, uh, we go talk to use a 2.5 costs acid three. And then basically what I vomited here is we would have to offer one to have appointment 5 to 2. Is it vibrating? So now way have all the information we need. We just plugged these enemy get 1.1 Flight three star. Finally, through this question, we would just compute all of these. Add them up one by one over four. Or is there a 40.25

Yuou Sun · Answer

okay. Problem. Someone just kept like ours, eh? What is D what he&#x27;s had. And what is that? Here&#x27;s the meat point rule, and fifth soil seems to rule, right? Easy going to an AIDS and port wine. And 10. So their acts is just, uh, 0.1. Right. And you can play the X one bar. All of them help them to get their tax ate the same. There have two. The same accidentally is one. The next one is, uh, 0.1 point 11 point to 113 extended to extend his too. Just calculate everything, Captain Calculate. And you can gather practice, sir. He is the CO two 1.5. They&#x27;re all 6361 Admit you go to one point by one, 8362 Andi, as is equal to one point by one by one night, that&#x27;s all

Chris Trentman · Answer

were given an integral and a value of n and in each part, whereas to use a particular rule, approximate this integral with this value of end and to round our answer to six decimal places. The integral is the integral from 0 to 2 of e to the X over one plus X squared DX in the value of n is 10. Now in part A were asked to use the trapezoid rule to use the trapezoid rule. Virtual wants to calculate the values of dysfunction at the boundaries of the intervals. Before we do this, we need to figure out what the intervals are. So the interval wit Delta X is going to be the end point to minus the other end 0.0.0 over end, which is 10 or 1/5 which as a decimal, this is point to and now we set up a table so the first row will be the values of X, and the second one will be the corresponding values of the function F. Now we&#x27;re going to start with X equals 0.0 and move all the way to X equals 2.0. In fact, maybe I shouldn&#x27;t do rose. I might want to do columns here because it&#x27;s going to be rather long. So we have X and F of X extort cells as 0.0. If you plug in 0.0 to the function, eat the X over one plus X squared. You get one. Since we want to round to six decimal places, I&#x27;m going to calculate the function out to six decimal places. You could get a more accurate answer if you calculate more decimal places. Next we have the X equals 0.2. So we add the step size plugging in 0.2 F of X is approximately 1.1744 to 6, and so on. Next we add two. We get four, for which X is approximately 0.4 and FX is about 1.286056 We continue in this manner, increasing the value of X by 0.2 at each step and finding the value of ffx to six decimal places. I&#x27;ll let you do this on your own. So completely calculations. I get this table of values now you know, for the travels away rule that there are and plus one which is 11 terms in the some and the some, which will call the trapezoid Rule t 10. This is approximately are integral and exactly is besides a bar interval. Delta X, over to times air function evaluated at the 1st 0.0 plus two times function directed at the second point. X one plus two times the function evaluated at the third point, x two and so on, all the way up to two times a function of values It the 10th point x nine plus the function evaluated at 11 point x 10. And then, if you plug in the values from above to Delta X is too, and we have f of X zero throughout the next 10. On the right hand side here, you&#x27;ll find the some is approximately 2.66 0833 Now, in part B here asked to use the midpoint rule, so this is going to be similar to part A. Our interval with is still point to, and now, instead of calculating values of the function at the end points of the intervals, you want to calculate them at the mid points of the Internet. Hence the midpoint rule. I&#x27;m going to draw another table here. So our first calm will be X and the second column will be the value f of X. And so we start off with instead of exit 0.0. We start with the midpoint of 0.0 and 0.2, which is 0.1 then will increase by 0.2 all the way until we get to the midpoint of 1.9 and 2.0. I&#x27;m sorry. 1.8 and 2.0, which is 1.9 Sorry. This should be three 1.5 0.7 and so on. And just as in part a. Well, simply plug in these values of X into our function ffx, which is eat the X over one plus X squared. So, for example, if you plug in X equals 0.1 F of X is approximately 1.94229 and we can continue in this manner. And so we obtain this cable of values, starting with 1.9 4229 and ending with 1.450302 At this point, I&#x27;m going to erase some of the results from part A to make room for those for part B. So, um, because we&#x27;re using the midpoint rule, we know that the number of terms instead of being end plus one, is going to be only and minus two or a total of eight terms. I&#x27;m sorry, should be a total of end turning. So we have a total of 10 terms here and our approximation, which will call em 10. This is equal to formula is Delta X Times and then we simply have the some of our function values of these midpoint so f of x zero plus f of x one all the way up to effort X nine for these 10 turns and plugging in the values from the table into this, some in calculating using the fact that Delta X&#x27;s point to this song is approximately 2.664377 So that&#x27;s your answer for part B. Finally, in part C, we&#x27;re asked to use Simpson&#x27;s rule to approximate so using Simpson&#x27;s rule is similar to the previous parts. I&#x27;m going to override part a here, so Part C begins in the same way that part eh began the step. Size is going to be the same 0.2, and for Simpson&#x27;s rule, we&#x27;re going to use the same table from part. So we calculate the value at the interval boundaries. So we start off with X equals 0.0. We go all the way an instrument 0.22 x equals 2.0, and we see the F of X ranges starting at 1.0 and then our table will end with 1.477811 We simply plug in the value of X and to eat the X over one plus X square to get the right side of the table. And with these function values, recall that Simpson&#x27;s rule as yes and plus one or a total of 10 plus one or 11 terms, just like party had 11 terms. And here I&#x27;m going to erase the results from Part B to make room for the results from Part C. Now our approximation, which we&#x27;ll call s 10. It&#x27;s Simpson&#x27;s rule in the n equals 10. Well, this is in the formula. The steps sides Delta X, which we know is point to over three times and there&#x27;s some is f of the 1st 0.0 waas four times, half of the second point x one plus two times f of the third point, x two plus four times f of the fourth point x three. When we continue this pattern all the way until we get up to because two times ffx eight plus four times f of x nine and then plus just one times f of x 10. So are some does contain 11 terms And if you plug this result into your calculator and you use the results from our table, this son is approximately 26 decimal places. 2663 244

Problem

Solve each given problem by using the trapezoidal…

Problem 19 Easy Difficulty

Solve each given problem by using the trapezoidal rule.
$\int_{0}^{3} \frac{d x}{2 x+2}=\ln 2 .$ Approximate the value of the integral with $n=6 .$ Compare with $\ln 2.$

Answer

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