consider-the-integral-int_0pi-sin-x-d-x-a-use-a-calculator-program-to-find-the-trapezoidal-rule-appr

Question

Consider the integral $\int_{0}^{\pi} \sin x d x$

(a) Use a calculator program to find the Trapezoidal Rule approximations for n = 10, 100, and 1000.
(b) Record the errors with as many decimal places of accuracy as you can.
(c) What pattern do you see?
(d) Writing to Learn Explain how the error bound for $E_{T}$ accounts for the pattern.

Bobby Barnes · Accepted Answer

we want to use some kind of ocular program to help us evaluate the integral zero. The pie of sign of excusing the trump is over rule when it is able to 10 100 1000. And then we want to record the errors with as many decimal places as Acrisius can. And then, lastly, we want to see if we can find a pattern and then finally explain this pattern using the ever bone formula or the trump is idle method. Now, if you just have a calculator that can plug everything in, you don&#x27;t need to do everything that I&#x27;m about to show you how to do an XL. But in the case, you don&#x27;t others to inging about this. If you don&#x27;t have just a calculator that will do, trap is a little method right away. And one is by looking at this as a re mont. Some using this formula here and also recall that H is going to be B minus a over. Yeah, well, let&#x27;s go ahead and figure out what age should be in each case. So in the case of t 10 h is going to be over integrated from zero to pie, so it should be pi minus 0/10. We get high, Tim. So if we want to find t 10 this here is going to be the sum of So the pie over for pie tin over 20. Sum from I is equal to zero two in minus one, which is gonna be nine of. And unless for what? X I should be on. We can do that off on this side right here. Remember X. I should be eating too. A breakfast is in top right corner. So remember that X I should equal to a plus by times h But in this case, a is zero. So that goes away. And then we could just plug in H, which we found to be pi over 10. So this should be I pi over 10. Now, let&#x27;s plug this in for each of our values here. There&#x27;s gonna be Oh, aye, aye. Over too. And then plus, yeah, uh, I plus one times pi over Tim since this year is X I in this Syria&#x27;s Excite Plus one and effort bags this sign of X. Let me write that in green. So this here is f of X. Let&#x27;s go ahead and just replace AF here with Cy. That&#x27;s too big. So sign. Sorry. Now, if you were to plug this into a calculator that has remained some banksability, then that will give you the answer. Let&#x27;s go ahead and set up the other ones as well. And if you don&#x27;t, I can show you how you can do this using excel so we&#x27;ll just go through these little bit quicker. So in this case, it should be high. Over 100 which would give us t 100 is equal to. So it&#x27;s high over 100 divided by two soap I over 200 announced. Could be son from Isaac Go to zero up till 99 because I would be one minus in and then we would have signed, uh, I times pi over 100 and then plus I sign of I plus one high over 100. Because remember, we&#x27;re using this this formula right here and we&#x27;re just plugging in for H. And in each case, a is still zero. And so then, if you were to plug this one in to some kind of remind some calculated that would also give you tea 100. And then lastly, we&#x27;re Team 1000. Just give us pi minus zero over 1000. See the tubes, Opie. Over 1000. That gives us Team 1000 is equal to high over 2000. Now, this is son from Isaac zero 2 99 for 999. A sign of Ai pi over 1000 plus sign of I plus one high over 1000. So looking all those three and should give you the area or approximation of our area. So let&#x27;s go ahead and see what I got when I plugged all this into itself. So just kind of explain it for N is equal to 10. So that would be This is NBC 10 and is able to 100 is into 1000. This top part here was just helping us solve For what? Our value of h is, um, this column of ends here is just for placeholders. Um, just in case, if I would have plugged something in weird and it went past the value I was looking for, I at least know I needed to stop at it and see 10. The next column gives just our input values. So that would be like a plus h plus age plus age. Plus age is going all the way down like that. Ah, than F of X is just taking whatever value we had here, the sign of it. So that&#x27;s how we would get that calm of FX is and then to get area column What I did waas I added these two cells together, divided it by two. And then I just repeated that so like, this would be I is equal to zeros area. Then the next one here with the eyes that one all the way down to here I is equal to 10 and I just repeated that and then to get this total area, I just summed up this entire call him here. So that&#x27;s what I got. Uh, and then for any 0 to 100 I got this area the 1.9998 and then for any secret 2000 I got 1.999983 by five. And I just have these values down here for each just so if you want Oh, double check with what I did if you set this up in Excel that at least er it values should match Oppa&#x27;s Well, now they say we want to then in part, be figure out the error. And so we should probably say how we got this equation here, that the error is to minus Tien. Let&#x27;s go back over to here. We&#x27;ll figure out what this value of sign actually is. Remember, we would just go ahead and integrate it. And this is going to be co sign XVIII Wade from zero to pie buggin end. Hi would give us I&#x27;m sorry. That should be negative Coast. So plugging in pie would give us negative ones. We have negative one times one and then minus when we plug in one, which would be just negative one. So the negatives in each case cancel out and we end up with an area of two. So the actual area and B do, and then we just subtract it from what we got from each of these right here. So that&#x27;s how I got these three values. And they asked for us to find a pattern in the error and in our case, noticed that between each of these. There&#x27;s a multiple of 10 to the negative second cause. Remember that E is just saying times 10 to the power of whatever follows. So the pattern is 10 to the minus two multiple to get next, ever. And this is just when in increases by a factor of 10. Then they asked us to use the actual air about formula to describe this. All right, now what I&#x27;m going to do is I&#x27;m just going to collect these two things together because for all of ours, neither of those should change what I highlighted in blue. So I&#x27;m just gonna be be minus a over 1 80 I&#x27;m sorry, that should be over 12 1 80 is for Simpsons. So over 12. And I&#x27;ll fix it right here as well. Over 12. And then we have m the double prime Now, h remember, it&#x27;s supposed to be B minus a over in squared. Okay, so we can go ahead and move that b minus Say to be my to say, cubed over 12 in double prime, and then we just have one over end squared. And in each case, if we increased by a factor of 10 than end would increased by 100 says We&#x27;re plugging in 10 and it&#x27;s just getting square. To become a hundreds of this right here is why we would expect that pattern we saw in parts.

Yuou Sun · Answer

Okay. Problem 21 first part A. We wantto speculate. T turn on turn asked. Her function is quite easy. Right. So we can just use Ah, one minute here. He, um and asked, you know, everything in the formula. Just use a formula hand calculate for the correct answer. After technically we can I he is he go to one point by a 35 a mystical to 2.0 eight to and ask is you go to to a point 00 You know what? Well, as it&#x27;s quite bad, I see the best right part b o and we should find the corresponding Aris. We have a form of that here. I just copied it from the book. Ut Yeah, have? Yes, right? So we want to know the K. Okay, it&#x27;s ah, second drag kid. Oh, since the function is quite easy, we can&#x27;t know that the second directive is also side axe. But there&#x27;s a man. It&#x27;s about there is a negative, but it doesn&#x27;t matter. So Kay is always equal to what? And the force directive off. If it&#x27;s just, I ax So the the maximum age also just what we can use this. Uh, okay. Times high. Okay, let&#x27;s start. The T is equal to one times higher to the three. Ready by it. Wow. Oh, was the end I forgot. And it&#x27;s a good 10 right? And yeah, it&#x27;s just Hi. No. Oh, this is 24. Sorry. 24 times. And yes, it&#x27;s just hi to defy and divided by what&#x27;s coming again. Oh, 180 times. 10 to the fourth, right. E We want to calculate the act. The extra iris right? We can just in to grow This function is to off its opener. Sweet, right? Yes, Cy, Thanks. Hero to pie. He acts these people toe cause I wa minus costs I minus one, right? Oh, hi. Sorry. Oh, cause I empire minus zero Oh, cause and zero minutes causing pie. Sorry when those coast line pie. So it&#x27;s just too right because all all our solutions is equal to two. Right? So we can compute actual Aras at Dr Iris. Let me see. Ah, yeah, Yeah, I minus he two minus one point night, eh? 35 It&#x27;s just through here. Oh, I say what right. No problem. And yeah, um, privacy. Go toe to toe minus 0.0 a two. My best deal. Two on us, equal to minus 0000 What? Right. Remember, huh? How long do we have to choose? And so that a proclamation effort? We&#x27;ve seen one nowhere. While 1234 Bye. You can make a father hi to the hell. Yeah, I divide five Times Square is more quit then. Oh, how many? 05 right? And just Yes, the solve it. And he&#x27;s quicker than I o a well nigh. Am I Correct? Yeah, I&#x27;m cracked, huh? And the 2nd £1 of served wine. Just use a formula here, and we have Ah, your question. Just solve the caution. We can get the answer right?

Jiji Peter · Answer

for ricin 18 6 20 using the rule or integration from one to do yeah to the power one over X, d X and in equals 10. Does the X the woods off interval with equal to minus 1/10? It will go, went one. Then we conclude the values at the interval. Boundaries for F X equal. Mhm, the power one over X and each X. We will find the value if it&#x27;s at X equal one. Fix it going. Seven. Wow! And it&#x27;s equal 1.1. It takes equal 2.48 at X equal. One way to fix equal 2.3 and etcetera still X equals two. If X would be equal 1.64 Hey! And we right out of the refrigerator rule for in the plasma at our condition. 11 terms and they feel in the numbers of from the table through de 10. With equal. There&#x27;s the X for over to off the blind boy If exports plus who, if it&#x27;s one lost to If the accessory etcetera, then to if it&#x27;s night loss. If it&#x27;s again by director institution, we will find the value off teeth and equal 2.2 19 76 Using the midpoint toe, we calculate the values at the mid points off each interval. At the checks, we will get the value off you&#x27;re fixed, you will get the value of X equals 1.5 Sorry x equal Ongoing 05 to fix equal 2.59 at X equals 1.15 If a sequel don&#x27;t work 38 at X equal one going to five if X equal, Don&#x27;t worry total and etcetera till X equal 1.95 to fix equal one Brain 66 nine For the midpoint rule we multiplies of its off. Each interval tells the X Times is a some off each fx upon the midpoint table. So in with him it won the X off the Blind Boy if in one plus it&#x27;s a me too glass, if a mystery plus, if in For Plus it&#x27;s in my life, plus etcetera till a feminine so and within. With equal two point 09 102 for Section B, take the 1st and 2nd derivatives off FX because effort double dash X is decreasing on the interval, one to do. The maximum value occurs at the left. Interval pound 81 1. Yeah. This is the lower limit for King. So if slash x equal minus one over exist where the blind bite you is a power one over x the federal dash X equal toe over. Execute multiplied by It was a power one by not one over X plus one block plus three look dreary. Last one. So far X is about four multiplied. Boy e is about our one over X. By substitution, we find that if fallible dash at X equal to equal Oh, boy. 515 and the F at X equal one equal 8.154 And this is the value off K, Uschi A equal one and p equal to In the e t equations off. We will find 80 equal. Okay, but the blue eyed boy B minus a Yeah, que Well, what? Well, and Miss Squared my director institution. We&#x27;ll find that it is smaller than sorry. He teach smaller than particularly Bob Wind Oh, 68 And using Kale equal one and be equal to in the e m equations off. So he am smaller than or equal key about the blind boy. B minus e que Well, What During 34 in the square. 24. Any square so e m. Smaller than or equal or going for 34 four section C If X equals e to the power one over X so if finish x equal A to the power one over X off the blind boy minus X to the barber minus two. If global Dash X will equal he to the power one over X multiplied by two x plus one over experts of our four F global dish is decreasing and more than 04 so the maximum is at X equal. One key equal, incurable dish for X equal one equal three e about it going 15 for eight. But since then, the Bozidar by said that trapezoidal error bound expression to smaller than or equal over a who won. So we will find that K multiplied. B minus A. You will, sir 12 in the square, smaller than or equal or going toe one. So ST e the blood boy to minus one. Q. Over the wealth in this world smaller than or equal who went Oh, go on. So e over four in the square. Smaller than what we&#x27;re going toe one and so and with bigger them for equal. No. You for what went for for 44 About 82 wait for rounding up to the nearest integral. So in equal it is three bye said the midpoint. Error bound expression toe 2.1. So key applied by P minus Take you over 24 in the square. Well, smaller than equal. We&#x27;re goingto one by directive institution. We will find that in the square. Well, bigger than e over eight multiplied boy. Oh, fun. And then and then will bigger them or equal 58.3 and rounding up to the nearest integral. So in we&#x27;re equal 59.

Yuou Sun · Answer

Okay, I start. I want to find the t&#x27;s up. 10. I&#x27;m something. We know the function is this one and we know and is equal to 10 right? And the B and es We know everything. So just use this formula here he end. And over here, ama, with this formula, we can compute. He turned and just calculation, there&#x27;s no tricks here. Just calculate you have got to park out there. Here, R B, we need to have to make the arrest. Here&#x27;s a function here, here to function. So we want to know. Okay, were Kay&#x27;s. So we need to know the upper ball off the second directive. Right? Let&#x27;s start after is equal to eat to the one of our acts. Oh, sorry. Ah. Of prime is ego Thio, Uh, minus X square or one time? Speak to the and a prime prime is the go to X more Ah, maybe maybe in other ways. Quicker, right? Minus Thanks. There. Finest too. I need to the thus. Thanks times. Ah, minus X square times me to the ex. What? That&#x27;s because the directive, right? You prime We plus you re price right and is to e Thanks. Plus right. Plus, there&#x27;s a negative Negative, that&#x27;s for Yeah. I need to be right. We want to find the maxim off the front crime. So we take us our directive. We can off. Where&#x27;s needs me? That surgery is smaller than zero. Right? If we take a directive up this one and it becomes a negative what? Take a drag her off this once you become a negative, and this one is also connected. One so so certain Detective is smaller than Gerald. You cannot have next him. Hello. Oh, this is when that. But actually it&#x27;s equal to what? So the So the maximum is three. Right? When we know that K is equal to three, uh, use the formula K times once where a cube. They only bite Wow. Times 10 square. Where is he going? To written me more, right? And used a calculator. You can find that, uh, this is equal to 6.8 times. Tend to the captives free. They&#x27;re the three, right? And use a formula here. We can find a yam. It&#x27;s just 1/2 off the e t. My biggest fear is that 24 years, so yeah, it&#x27;s 3.4 times tend towards an active three. Here&#x27;s party Parsi. How long do we have to choose in such that they&#x27;re accurate to within? I wonder where that is. One weird 10,000 right? Yeah, It was a former gang three he derided by Wow, and Square is smaller than Yeah, Here&#x27;s the formula hand just technically. And there&#x27;s equal to the square here, then 83 and the 2nd 1 just. He raised his 12 and write a 44 here and just solved the potion equality. And they were heard of 59 that&#x27;s all.

Yuou Sun · Answer

problem. 25 for this problem, you should not. Is that he wants you to computer corresponding arrows. Now, that estimate estimate harassed you can use this. Asked with arrows. Ira Bond. Right. But the corresponding arrest it needs to It needs the I might just the same because, like Al or are Or, um or teeth just minus minus. Then you can find Yell yard. You can yet, right? I don&#x27;t know. And you need to know how to compute our Artie and PNM. You can use the here, the formula here and how its left hand side by our is the right hand side. We have ah formula here. Ah, here&#x27;s here&#x27;s Al How is equal to this one? Ours even knew this one. Just confused. There&#x27;s no tricks here. Just compute for ending for any single 5 10 or and 20 computer attacks future tax here. And just all the same thing is no one. So compute those things. And of course, Bonnie era. You need to compute I how to compute. I hear, uh, you can find a You find a bunking from 0 to 1. Yeah, it is equal toe X minus one that guys That&#x27;s because the directive off dysfunction is equal to that one. That&#x27;s a new tin likeness formula, right? If we go to Daryl minus minus, one is just one. So the eyes one and you can compute this one. But just about I cast the Hera is smaller because you created the interval into more small intervals. So there are approximately is more close to the real value I

Yuou Sun · Answer

problem. 26. I want to compute our Artie and and or unethical to 5 10 honey and to compute the corresponding harassed. First we need to know how to compute our Archie and am here the formula. There&#x27;s a book. This is Al. Oh, it is just Ah, left hand side. Did this one is now left endpoint approximation on are the right hand side. Yeah, right. It&#x27;s the sunrise here for each and you need to compute their toe acts and effects I for tea and am Here is a function of teeth. Here&#x27;s a moment off. Just compute. There&#x27;s no other ways too. There&#x27;s no tricks here. Just compute. You know everything. Just control And you don&#x27;t know your equal. No. Sure, your answer is cracked. Just just compared them with the real answer because you can compute the eye. Exactly. Right. Eyes equal to 12 x square. Yes. Right. And these people to my sex What? 21? That&#x27;s Ah, Newton Nablus. Right, because the directive off dysfunction is just six function, So it&#x27;s just, uh, minus Woz. Uh, sorry. One minus one or two. This huh? Your answer. If your approximation is close to one, huh? Then you&#x27;re on crack. Just just calculate them and yell yarn itchy! And I am just It&#x27;s equal to ay minus he or are or I&#x27;ll write because calculate, there&#x27;s no other way.

Problem

(Continuation of Exercise 27$)$ Repeat Exercise 2…

Problem 27 Hard Difficulty

Answer

a) $T_{10} \approx 1.983523538, T_{100} \approx 1.99983504$ and $T_{1000} \approx 1.999998355$
b) 1000$\quad 1.645 \times 10^{-6}$
c) $\left|E_{T_{10 n}}\right| \approx 10^{-2}\left|E_{T_{n}}\right|$
d) $\leq 10^{-2} \cdot\left|E_{T_{n}}\right|$

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a) $T_{10} \approx 1.983523538, T_{100} \approx 1.99983504$ and $T_{1000} \approx 1.999998355$b) 1000$\quad 1.645 \times 10^{-6}$c) $\left|E_{T_{10 n}}\right| \approx 10^{-2}\left|E_{T_{n}}\right|$d) $\leq 10^{-2} \cdot\left|E_{T_{n}}\right|$

Calculus; Graphical, Numerical, Algebraic

Kayleah T.

Kristen K.

Michael J.

Joseph L.

Precalculus Review - Intro

Functions on the Real Line - Overview

Ross L. Finney, Franklin D. Demana, Bet K. Waits, Daniel KennedyCalculus; Graphical, Numerical, Algebraic

Catherine R.

Kristen K.

Samuel H.

Michael J.

a) $T_{10} \approx 1.983523538, T_{100} \approx 1.99983504$ and $T_{1000} \approx 1.999998355$
b) 1000$\quad 1.645 \times 10^{-6}$
c) $\left|E_{T_{10 n}}\right| \approx 10^{-2}\left|E_{T_{n}}\right|$
d) $\leq 10^{-2} \cdot\left|E_{T_{n}}\right|$

Ross L. Finney, Franklin D. Demana, Bet K. Waits, Daniel Kennedy

Calculus; Graphical, Numerical, Algebraic