solve-the-given-problems-using-simpsons-rule-exercises-15-and-16-are-the-same-as-18-and-19-of-sect-4

Question

Solve the given problems, using Simpson's rule. Exercises 15 and 16 are the same as 18 and 19 of Section 25.5
The average value of the electric current $i_{\mathrm{av}}$ (in $\mathrm{A}$ ) in a circuit for the first $4 \mathrm{s}$ is $i_{\mathrm{av}}=\frac{1}{4} \int_{0}^{4}\left(4 t-t^{2}\right)^{0.2} d t .$ Find $i_{\mathrm{av}}$ with $n=10$.

Gideon Idumah · Accepted Answer

we start by first finding the exact value off Integral from suit to three off. Sue over. X squared DX. This is not involved. The integral of one of our X squared is minus one of our ex. That&#x27;s minus two over X, violated from suit to three, which is nothing. Boats. Oh, if you plug that into a copy until you have zero points. 3333333 Now using the truck present, Our rule I have That&#x27;s my doc. Might would end accord to four on Dr X, which is a ghost to be minus a over. And it goes to my bees three, my ace to develop before that&#x27;s the cost of one of the four This implies That&#x27;s my X zero, which is a cause that sue my ex won&#x27;t will be two plus one over four, which is 94 4 My extra will be nine of our four plus one over four were just turned over four, which is of course, the five of it. Soon my exterior will be five itsu plus a wonderful four, which is a level of a four under my ex for you called 23 So Trump is it ever gives me and B minus a over to end, which is a ghost. Tow one of our four witches, demons over end demanded by two. That&#x27;s one of the aides they want to blind by my F Off X zero is to over two squared, then close about opposite area off, so multiplied by f X one that&#x27;s to over nine of our four school close. So if off Ex warned us to over five of its two squared plus so F of X terry, that&#x27;s two over 11 over four squared and then plus F of x one F of X, for which just suit over three squared. This is approximately equal to 0.3352 Now he&#x27;s in The Simpsons Rule, which is B minus a over three end, which is a Coastal B minus over and is one of our four divided by three. That&#x27;s one of the 12. So that&#x27;s one of the 12. The more supplied by my f of X zero is to over two squared because of senses rule. We are four multiplied by my f of x one. Is this plus suit multiplied by my f of x two Is this close fool? My effort, ext. Three is days and then plus f of X four blood visions of cock a little. You have 0.3334

Angela Guo · Answer

were asked Approximate Doesn&#x27;t his role here with the trapezoidal cool. So before we do that, you need to fill out the table on the right side of the table on the left. So are Delta X ISS being minus a over, and so, in this question, are a is negative. One R B is three. In the end, it&#x27;s four were dealt. Eggs is just one. That means we will start a negative fine. And we go up by one. There are 123 and I was excited. For each of these excise will be one over two three over five, really before one and hurry over to. Do you find these values by just putting your ex I into the back. Such is here. So now, with all these values, we can put them into our formula, for the trap is little. Well, part ay says integral. From 1 to 3 of three over five minus X, the X can be approximated by be minus a over end, which was calculated up here. That&#x27;s just one times that&#x27;s half time us through over Size was three over for plus one, and the last term is half of that so half of three over two. And if you plugged in and your calculator, you&#x27;ll get this is 67 over 20 hate Now Part B. We do the same thing, but this time you seem so as well. So that says into little from negative 123 of free over five minus X yeah, can be approximated by be minus a over Fri end will be Mine is a S 43 times on a 12 and that&#x27;s times what have plus four times free over five plus two times three over four, four times one last 13 over two. Well, if you put this injury calculator, you&#x27;ll get it is equal to 33 over in the last part. Pricey. Or ask the justice I await the integral No, that&#x27;s from negative. 123 five minus x d. X. So the three on top is just a constant, which means they can actually just pull that out. That&#x27;s three times integral of negative 123 of one over five, minus X d x and the anti derivative of won over five minus X is just why five minus X, but because they&#x27;re the negative. We have to account for that. Which means they&#x27;re gonna multiply thy negative one. And that&#x27;s a relative from negative one three. We&#x27;ll just move the negative, the front. So that&#x27;s negative. Three. And that would be one of five minus three. So that&#x27;s long, too. Line us on a five minus negative one of Lawn six and using our log rule. We know that one of two minus line of sixes just long of 1/3 and using the other law girl that tells us we can put this constant up as a power that&#x27;s long of won over. Three. The power of Negative three, which is just one of 27. So that&#x27;s your final answer. Um, if you want, you can put it into your calculator. That is just awfully. Three point 296

Shuyang Fu · Answer

Hi, guys. This problem is too. Approximate the integral from 0 to 4 for function square Rudolph X times E to the X so we can see the left and point off the interval. Zero in the right hand. Right in point off the interval is four and the number off sub intervals in is also four. So that means we need to divide the whole interval from 0 to 4 to four parts. Even me. So the left endpoint sex Not since he wrote the next note, X one must be wanting and then two in the three. And the last note is X four. So we have all these sub intervals with lens one. Okay, so now we can use these notes to evaluate this integral by using chaps Oi! The rule. So the integral from 0 to 4 square root off axe you to the ex the ex We will be approximately equal to be minus a So it&#x27;s foreign minuses Aeriel over to end. So it&#x27;s two times four multiply f off ex Not that is pluck ex, not to the function. So it&#x27;s zero times you 20 pull us two times, plug one to this function, so it&#x27;s square root of one times E to the one plus two times F off too. So it&#x27;s square root off to Times Square and then to times square root off three times e to the furt into the lost. The term has the coefficient toe one. So it&#x27;s just one times square little four times in 24. Okay, so this number, we can use the calculator too. Develop it. This number it is 102 point 5553 Okay, so the next thing is to evaluate to the same integral by the same since rule. So it is approximately be months, say, over three end. So it&#x27;s a four minus zero over three times before and then f off. Ex not is still zero for us four times square root off one e 21 plus two times square it off to the square. Perhaps the four again square it off three e to the third, and then the last coefficient is one. So it&#x27;s a one time so four times you to the for square root of four times 24. Okay, so this number is approximately 93 point 3752 Okay, So after we compute the chapel or the rule and Simpsons rule, we need to compare that these results with the true integral value, which we can get these value by using the A graph utility like a graphical calculator. So you for you input this integral to the graph calculator. We can see it is approximately 92 0.7 for 37 which was very, very close to the same sense of rule.

Shuyang Fu · Answer

Hi, you guys. This problem is too approximate and to grow from 0 to 2 for function square roots off one plus X cube so we can see the right end points off for the interval. Be close to two. The left endpoints A is zero in the end, the number off sub intervals It&#x27;s four. So we just need divide the interval from 0 to 2 to four parts evenly. Okay, so the first note is the left and 0.0. So it&#x27;s x. The XL one is 1/2 x two is one that three with three halfs x four is the right end 40.2. Okay, so we&#x27;re gonna use the chaps. Oi! The rule first. So just a plug or the note to the chaps Oi! The rules formula. So it&#x27;s to mind this is Ariel over two in which is two times before times f off ex, not so black zero to the function gets It&#x27;s square root off one plus zero cubed plus to times square roots off like 1/2 to the function we get. It&#x27;s one plus what half cubed and then to times square root off one plus one cube and then to times square root off one pass, three halfs cubed and then the last turn description one. So it&#x27;s square root off one plus two so we can use the calculator. Two. Calculate the value off this formula and it is approximately 3.2833 for the same sense rule. The only difference is the coefficient off the function value. Is that the notes? So we just plug or the nose to the formula here. So the beast to a zero and two times three times in, which is three times four and the f off ex not It&#x27;s the same. So it&#x27;s square it off one plus they&#x27;re repute and then four times f off axel one. So it&#x27;s square root off one plus a 1/2 cubed And then to times square with off one plus one cubed and then four again Square was off one plus three halves cubed. And then for the Lhasa Terry, the coefficient is just a once a square root off one plus two. Cute. Okay, so this number is approximately three points. 23 96 Okay, so two get so the chaps or the rule and seem since rule are two approximation off the chew value off the integral. So, in order to develop into the true value off the integral, we can use the graph utility like graphical calculator. So if you use a graphical calculator, you can get the graph the area below the graph. And you think that&#x27;s area too evil it this integral. That is approximately 3.24 one three. Okay, so we can see the same sense rule gets, uh, closer result through the two integral value.

Roman Frolov · Answer

So this problem want to approximately where if you definitely jewel using two different. That&#x27;s 1st 1 of the trip is over the rural 2nd 1 Simpson role. So let&#x27;s start it straps on the rule. So let me remind you what it is, if you can. A definite attentional from in to me. Oh, after a fax. Yes. Then it can be approximated a psalm with respect to Kane from one toe end off half with X K minus one plus half of pax Kate, you wanted by two times don&#x27;t escape. So in our case, half of facts is given ass Juan or one plus X squared. Hey, left and point of integration in general is equal to zero and the right on point B equals to one. They want us to use a number of something true and equals before. So that means that for any king Delta X keen, it&#x27;s gonna be B minus me, you wanting my end, which is in our case, one or sort. So when we know that we can compute act Well, it&#x27;s off. Actually, Rex Monix two extra erratic and explore the ones we&#x27;re gonna need. Two years and four months in. The drop is over there, the 1st 1 at zero Jazz A laughed at a point e, which is Europe X one is gonna be this scheme. Applause Delta X Plus one over four. So it is just one or four. That&#x27;s too yes, zero plus two days, one ofour, which is tour for or simply one or two. It&#x27;s the same thing. Can&#x27;t compute extreme similar weight. It is three or four and finally asks for It&#x27;s just won the right in point. So to use a formula right here, the trip is over. The rule. This one. We need the whale use of the function at all of this points. So last time get exactly this we take Axiron substituted into the function F, which is one on one clause export. So after Frank zero becomes one, want applause, You&#x27;re squared, which is just one after fats. One is one or one plus one over four squared. It&#x27;s a bit more complicated, but after a bit of simplifications will yet 16 or 17 after facts, too. Camps one over one plus one or two squared, which is one off work again. After bitter simplifications, we get four or five than half of extreme? Nope. Sorry of Act three chalice one or one plus. Now I know where 16 goes through your force. Words is nine over 16 simplified. We get 16 over 25 and finally is right. Dr. This point acts half of X for ISS Juan or want. Plus, once gorge, which is one, it&#x27;s one or two. So now we have all the components to compute approximations, trapezoidal rule. So let me open your shit, and we just substitute this valiant effort. Facts. Here I have X want aftereffects to half of Act three and 1/2 effects. Four. And well, I doubt the axe into this formula so we get it are, in general from 0 to 1. Well, Jax over one plus x squared. He&#x27;s approximately equal. It&#x27;s not exactly it&#x27;s approximately equal to Delta acts or two gonna fucked around this one or two. And inside, if me corruption it everything it through a sig meditation, we get F off AG zero plus two terms. Half of x one plus two James half of axe too. Must too James, half of Act three close, half of packs for so it it was too, on our death, actually. One of our force. So in front of here, one or eight and then we substitute the computer analysis of the function. We have one plus two times 6 10 over 17 plus two turns four or five plus through time, 16 over 25 plus one or two. So I&#x27;m not gonna torture you. Reserve methods, came the calculator over. Spend a bit of time figuring it out. But as a result, yes, uh, Ingestible form zero point. So on eight? No, to Salome 94. And so once on something so long. So it has infinite number of this most decimal places. So if I approximate it up to four decimal points, I get zero. If I rounded up up to a four decimal decimal places again 0.78 28 so that the approximation of this internal using the truck is idle, its nose exact value, and we can use different former for approximations. It seems, Admiral, we basically have all the components to do that. So, using a Simpson rule, I will end too. Girl from zero to Juan. Well, half of acts deaniacs gonna be approximately equal to Delta Axe or three? Yes. Ah, Different message off. Approximately into growth. Uh, half of acts not here. Plus four. Just half of x one Kloss two There&#x27;s four x two glass for there&#x27;s, uh, off extreme Plus it&#x27;s for So we have all the conformance already. So we already computed x zero x one x two extra explore There is the same as for this trip is a literal with to use and some controls So why no delta access to one before? So in front we hear one or 12 and inside of one clause four times 16 or 17 plus two times four or five plus four times 16 or 25 plus one or two so again was only torturing you is our methods I tell you that if you use a calculator or do we just sat on my hands? You should get again in a decimal forms You&#x27;re a 0.78 five stream 92 again with a number of points. If I round it off, do four decimal places I have Sorry someday problems zero point selling eight Final four s o. The result is quite close to where a God with trapeze over the rule, but a bit different because again, it&#x27;s not exact. Well, yours a integral it&#x27;s an approximation, and different methods of approximation can can give you different results. Uh, so if you want to know actual results, I&#x27;m just out of curiosity actual well using to go from 0 to 1 off one or one plus export the axe. Yes, zero point selling eat for three. Nine. Eat one. So it&#x27;s very close, in particular to the Simpson rule. Get the number of points, so it&#x27;s approximately zero point. So in two games four. So if you run it up, we get the same result. But they&#x27;re still slides of different so many ways you can approximate introduce. You can also use the Riemann sums if you want. I feel free to use any graphing calculator to do that. What Here it is for this problem, we computed to approximation saying, That&#x27;s it

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Basic Technical Mathematics with Calculus

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Basic Technical Mathematics with Calculus

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

Grace H.

Catherine R.

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus