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Problem(Elements of Numerical Integration) Apply the Trapezoid and SimpSOn Rule to approximateIn $ &xand find &n upper bound for the erTOr in the YOur appro...

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Problem(Elements of Numerical Integration) Apply the Trapezoid and SimpSOn Rule to approximateIn $ &xand find &n upper bound for the erTOr in the YOur approximation_Problem(Composite numerical integration) ApproxiIate using 4 subintervals in (a) Composite Trapezoidal Rule posite Midpoint Rule; (c) Composite Simnpson Rule.Com -

Problem (Elements of Numerical Integration) Apply the Trapezoid and SimpSOn Rule to approximate In $ &x and find &n upper bound for the erTOr in the YOur approximation_ Problem (Composite numerical integration) ApproxiIate using 4 subintervals in (a) Composite Trapezoidal Rule posite Midpoint Rule; (c) Composite Simnpson Rule. Com -



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Trapezoid Rule and Simpson's Rule Consider the following integrals and the given values of $n .$ a. Find the Trapezoid Rule approximations to the integral using $n$ and $2 n$ subintervals. b. Find the Simpson's Rule approximation to the integral using $2 n$ subintervals. It is easiest to obtain Simpson's Rule approximations from the Trapezoid Rule approximations, as in Example $8 .$ c. Compute the absolute errors in the Trapezoid Rule and Simpson's Rule with $2 n$ subintervals. $$\int_{1}^{e} \frac{d x}{x} ; n=50$$

Alright from 48. So using the chaps are drilling Simpson's rule because of the following integral roles. So we have the function f of X is gonna one over X squared plus one and with the integral from zero to 0.78 54 or five or four and then we're gonna use 64 subdivisions and 128. So starting with so the fact exact value is supposed to be 0.66 5774 Thio successful places for an equal 64. That gives us Delta X ah 0.10 point 01 2272 and we could find the traps. What approximation? 0.665766 Using an Nichols 1 28 can get a better approximation. Go to 0.665772 We'll use Simpson's formula, which is just four times the trap side approximation at 1 28 minus the chop shops with approximation. At 64 you get 0.665774 and now we confined error for the traps. OId we get the sixth decimal places 0.0 002 An air for Simpson's We got AH, 0.0 zero point and then six zeros.

In this problem, we have the definite integral from one to e of one over X and an equal 50. You are a weird, calculated trumpster rule approximations to the integral. He's in N and two ends of intervals that he's 50 and 100 of intervals a Barbie. We will find Simpson rule approximation to the integral, using one hundreds of in tools. And in Percy, we will find the absolute errors in the Trumpster rule and seems to rule with one hundreds of intervals. So in all cases, we're gonna use mad lab implementations of trouble, so rule and seems to rule to calculate the approximations and is in those implementations. We found that the traps a rule approximation to the integral using and serve Interval studies 50 serve intervals. These are approximately equal to one point 0000 85 C or nine then and the Travis a role using two pence of interval cities, one hundreds of intervals. He's approximately equal to 1.0 2127 Now I'm Barbie. We're going to calculate seem several approximation to the inter girl using one hundreds of interns. For that, we're going to use the relationship between the Simpson role on Travis. I'd rule given in the textbook, and that's the following. Guess of to dance of intervals. Simpson will for two and serve intervals is equal to four times the Travis A rule approximation using to answer of intervals minus the Trump side rule approximation using and serve intervals over three. So for any holes 50 we found that IHS of 60 Sorry, it's a 100 central rule Using one hundreds of intervals is equal to four times Travis A rule using one hundreds of intervals minus Travis Rule is in fifties of intelligence over three. So seems to rule approximation. Using one hundreds of intervals is equal to. In fact, it's ever seemingly too. Four times your approximation to t of 130. Steve's value here 1.0 2127 finest T 0 51 here that is 1.8509 and all that over three and use a calculator. We find that it seems to rule approximation to the interval is in 170. It'll separate timidly, but the syrup point I 99 999 997 And in part Sina, we're going to calculate the absolute errors, using both methods ever so rule and since rule using when hundreds of intervals. And for that we're gonna copulate first. The except into role in this case is very easy to calculate. We know that these Inderal here's you go to the natural ovary them of the answer the value of X you already between X equal one on X equal be And that's equal to the natural rhythm of e minus the natural rhythm off one which is one minus you, which is one. So the central issue go to one. And so the absolute error into traps a rule you've seen one hundreds of intervals is equal to Yeah, So the value of one minus t of 100 that is his father here to find that this is one minus one point. T. Rowe, 000 2127. And that's equal to 2.1 27 17 times 10 to the negative. Um five. In the absolute dearer in seems a rule using one hundreds of in devotes is equal to yes, a value of one minus T. C s of 100 cities. The's approximation here. So yes, everyone minus zero point 99 99 99997 And that's approximately 123 point serial times 10 to the negative night. So we have found much better. Approximation isn't since a rule using one hundreds of interval than to travel through using the same number off 70 balls.

In this room. We have a definite integral from Syria to buy fourth of 1/1 plus X Square in any single to 64 you are a We're going to calculate the traps a rule approximations with ease into rule using end to end serve Interval Studies use in 64 1 28 seven titles. Barbie. We're going to find The Simpsons rule approximation to this interrelation two ends of intervals. And for that we're going to use the relationship between the Simpson rule for to answer intervals on the trip. So for an and to insert intervals Parsi, we will compute Travis a rule and seems to rule. Absolute errors is into ends of intervals. So in Ah, in this problem, we're gonna use Trappist several implementation computer implementation in Madalla to calculate the approximations to the integral using and two times and serve intervals. We do that. We're going to find that the trumps a rule approximation to the interval using 64 serve intervals, he said recently, equal to see your point 66 57 66 21 and the trouble So rule approximation to the interview RL using 128th of intervals, which is two times in it's approximately equal to syrah point 66 Okay, 577 186 So this rd two approximations using Travis a rule with n and to answer intervals in Part B. We're going to use the relationship between the Simpson rule for, uh, using two ends of intervals and the Travis A rule using an and two instrumentals. We know that IHS of two times in is equal to four times t off two times in minus T of in over three. It means that The Simpsons rule approximation to the internal use in 128. Seven Devils is equal to four times the traps, a rule used in 128th of intervals minus Travis, A rule used in 64 So intervals over three. Now using the values calculated in part A. We find that seems to rule approximation to the integral. Using 128th of intervals December timidly equal to four times T of 128 that he's he's following here. Sorry, he's value here. This one we get that. Ah, we have four times 0.66 577 one Aide six minus t of 64 which this is done here. See your 0.665766 21 over three. And thats approximately equal to I see your point 66 577 uh, 374 That is the Simpsons rule approximation to the integral use in 120 years of intervals. Eats approximately equal to syrup 0.66 57 374 And if we use an implementation of the Simpsons rule Ah, on a computer, we shall find the same result if we calculate A S of one on 28 directly directly in the computer. So in part C, we're going to calculate the absolute errors for both Travis A rule. And it seems to rule using 100 training, training serve intervals. So first, we're going to calculate the exact value of the integral that is dangerous from zero to by fools of 1/1 plus x square. It's equal to another. The primitive is able to the in verses attention function. There is the our attention function off bigs. You awaited into zero on by fools in that sequel to We Are Tension Tough by fourth, minus the Ark Tensions Off Zero This is zero. So the sequel to the Ark Tension tough by fourth And that's upper seem Italy equal. We gotta use an approximation here. See your point 66 five 77 375 So we're going to use thes approximation as the detect value. You know that it is. No, it's not the exact value, but we had to use some number. Oh, are, uh, on approximation. Very accurate to these, Integral To calculate the absolute error. So the absolute terror for the trip side rule he was in 128 7 devils is equal to the absolute value off separate seeming legal because we were going to use the exact value, which is in fact on approximation. So it's zero point 66577 375 minus the visceral approximation to the integral. Using 100 journeys of interval cities, these are here. Hey, I see your point. I see your point 66 577 one paid six and thats approximately equal to 1.89 time. Stand to the negative six and the absolute error for the Simpson rule. You seen 120 serve. Interval sees have seen it legal to the absolute values here, a 1200.66 577375 minus three. The same for real approximation to the integral. Using 128 with the will studies. He's out here. So we had 0.66 577 374 which is, um, approximately good to want for one times tend to inactivate, so we have a much better approximation is in the center rule of ah than the Travis's rule used in in both cases, 128 7 titles.

Alright from 46 for the function. F Mexico's actually fourth equals zero because to and ends of 30 and 60. So when you use the traps one method first, starting with the step size ah, 0.6667 That's two divided by 30 Get trap side approximation of 6.411 I'm an equal 60. That's a Delta X 0.33 So half the size is before we get S a t ah, 0.6403 Then when we use Simpsons for step size with 60 I guess 6.4. Then air in the trap side method. Absolute value of E Minus T, which can re 0.296 and Air and Simpsons going to be 4.994 times 10


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