Doing Approximate Integration Calc II

追加: 2026-05-26

[00:00:02]And we are all back in another video.

[00:00:06]We're back in another live stream.

[00:00:09]Crazy. We're doing like daily live streams now. I don't know how we're doing that, but we're doing it. Maybe because we have a lot of homework. And of course, the number one the well, excuse me, the number one question you all must have for me is how much did I enjoy my long poop yesterday? because I believe I did mention that in my last live stream and I enjoyed it pretty well. Thanks for asking. Anyways, today we are doing a new concept in calc 2.

[00:00:43]Yesterday we saw partial fractions.

[00:00:46]Barely finished it. We stayed up like till 1210 a.m. just doing some coding because we also have other classes uh not just this one. Hey, in today's video, in fact, let me share my screen.

[00:01:06]In today's video, we have even more to do.

[00:01:10]So, I shared my screen so I can show y'all what I have left for this module.

[00:01:14]This is an accelerated course. If in case you guys did not know, it's six weeks long and there are four modules.

[00:01:23]So it's a week and a half per module. We did this last Monday. Today's Monday. So we did this a week ago. The substitution rule was done on Tuesday by us. We did this on Wednesday. This took us three days and we completed it on Saturday. It was a very tough a very tough concept.

[00:01:41]We did partial fractions yesterday and today we are doing approximate integrals. Tomorrow and this is due for Wednesday. Tomorrow we do improper integral is also due for Wednesday. and our review for test one and our test one is due on Thursday. Then we begin module two which we will see some applications.

[00:02:02]I'm not sure if it's going to be easier.

[00:02:03]The applications of derivatives on for calc one was much harder than just derivatives but but again integrals are much harder than derivatives. Who knows the applications are going to be easy.

[00:02:14]Hopefully they are. Hopefully they are.

[00:02:16]Hopefully. Then we have some series. I heard these are some of these are tough like tailaylor series or power series whatever conversion test convergence tests and then we have parametric and polar curves well hopefully that goes well six weeks let's do this it's it's nearly an assignment per week in assignment per day my bad so it gets tough like that and hold on I don't think I left my camera like that right there oh if I move it there looks closer If I put it there, it looks farther.

[00:02:53]You know, I think that's okay, right?

[00:02:57]Think y'all can see me clearly. Oh, no, no, no. Hold on. Actually, I wanted to test this out. If we put this uh There we go. Y'all will see me do work like this.

[00:03:18]That's how y'all will see me do work.

[00:03:21]It's actually not bad, is it? Because if I take it off and put it around there, you see me like that. But here, it's totally different. I think it's more sigma if you ask me. So, that's what we will be doing. You can see my iPad right there. And yeah, you'll just see me work. Well, uh, since no one is watching, we will begin now because there's nothing left to talk about since you guys aren't there. So, this video luckily was only 16 minutes long.

[00:03:55]We watched the first 12 and we covered the left, the right, the midpoint, and the trapezoid rule for Ryman integration. It was it was cool. And by the way, I I already saw these first three, left, right, and midpoint. We I just saw the trapezoid, but I mean I think I'm getting it overall. We have one more to see which looks like he'll cover in the last four minutes. Simpsons rule, but I think it's we have Oh, we need to know the Simpsons rule. So, I guess we could watch this quickly.

[00:04:28]Let's Let's do this. So, just going to begin taking notes over here.

[00:04:36]Um, Simpsons rule. Okay. On each consecutive pair of intervals, the curve y = f ofx is approximated by a parabola. Interesting.

[00:05:00]On each consecutive pair of intervals.

[00:05:03]Oh, so that probably explains this diagram right here.

[00:05:14]I see. Okay.

[00:05:16]So, do they use a parabola to estimate an area or do they estimate the area of a parabola? I'm not sure.

[00:05:24]Let's see.

[00:05:28]In fact, I'm just going to open. Oh, wait. Already have it right there.

[00:05:34]Yeah, we saw that. We saw that. Very easy. Not bad.

[00:05:43]Uh, okay. So, n must be even. Okay.

[00:05:47]Let y sub i = f of x subi.

[00:05:55]Okay.

[00:05:57]Doesn't look too complicated.

[00:05:59]The integral using the Simpsons rule equals delta x over 3 times y sub 0 + 4 * y sub 1 + 2 * y sub 2 + 4 2 k 4 2 Okay, the pattern 1 42 42 42 42 41 Okay.

[00:06:35]By a parabola. Interesting. So uh CR was I going to write curve curve y equals f ofx X approximated approximated by parabola.

[00:07:23]I spelled out PA parabola.

[00:07:30]Okay, I see. I see y'all.

[00:07:47]>> Gotcha.

[00:08:04]Interesting. So we have x sub0, x sub 1, x sub2, xub3, xub4, xub 5, xub6, xub7, xub8.

[00:08:16]So 2 1 4. Wasn't that the pattern? All right. Well, clearly we go up. Oh, no, no, no. Four, two, four, two, four, two, four, two, four.

[00:08:41]Okay, let's write that formula down.

[00:08:43]Hopefully, we understand where it comes from.

[00:08:47]Oh. Oh, okay.

[00:08:54]And is three just n or is three like always?

[00:09:00]Right.

[00:09:03]From A to B, f ofx dx equals I see. Okay.

[00:10:16]Yeah, here we have the four and then the four.

[00:10:22]Okay, gotcha.

[00:11:23]Okay. So first uh nothing multiplies or one multiplies f ofx sub0. Then we have 4 * oops dude no f of x sub 1 + 2 * f ofx sub 2 plus 4 * f ofx sub3 plus and oh and f of x sub4 is the last point. So f of x sub4 close. Mhm.

[00:12:25]Yeah.

[00:12:34]Okay.

[00:12:50]Mhm.

[00:12:59]Mhm.

[00:13:28]Oh yeah, look at the trapezoid becomes much more accurate. The middle point also becomes pretty accurate. The right and the left don't really become as accurate, unless I'm reading this wrong.

[00:13:40]And the Simpson rule is the most accurate.

[00:13:50]Right. Uhhuh.

[00:13:58]Yeah. Uhhuh.

[00:14:02]Like 10,000.

[00:14:05]So Simpson is the most accurate far.

[00:14:17]I think this is important.

[00:14:23]Yeah. In general, when n multiplied by k here for left and right mold.

[00:15:19]decreases by factor of K.

[00:15:29]The air for the trapezoid and midpoint rule decrease decreases by uh by factor of k squ error.

[00:16:12]I had a feeling that they're going to ask us this in the homework, so we should just write it down.

[00:16:34]Awesome.

[00:16:36]Very cool. Very very cool.

[00:16:40]So that is the entire video. So we should have no doubts within the homework. We should have no doubts. We should not even struggle. This does not even seem that bad. I don't see how this can become bad. I don't. But let's do it. Oh, great. We have 21 problems. Oh, wow. That's a lot of problems.

[00:17:05]Looks like the key for this is to go quick here. So, I'm going to take a picture.

[00:17:27]This midpoint rule the trapezoid rule.

[00:17:59]Very good.

[00:18:01]And finally, Simpsons rule.

[00:18:10]All right, we have these pictures. There is absolutely no reason for us to struggle as we have done in the last ones and let's take a picture of the example right because why not to go extra safe now I am going to split my screen so I'm going to do oh um how do you split the screen all right I'm going to go to photos Oh, I remember that. That was pretty cool.

[00:18:56]Where did these fors come from, bro?

[00:19:02]What's this, bro? Like, old photos, whatever.

[00:19:13]So, let's split this.

[00:19:17]Let's put this up here.

[00:19:21]And let's take good notes.

[00:19:43]Okay, here we go. It is time to lock in.

[00:19:50]There is no reason to struggle right here. It is time to lock in. I'm telling you, guys. I'm telling you.

[00:19:57]It is 12:13, huh? Yeah. I'm also wondering to myself, it's also pretty late. The thing is that since I stayed up till 12:015, what? 12:15 yesterday?

[00:20:11]Um, I believe I woke up today like at 6:50.

[00:20:16]Then I fell back to sleep and then I woke up again at like 8:30.

[00:20:23]I did my devotional around 8:15 and then I think after that what did we do? I just hung out with my brother, played with the dog, ate breakfast and then around 9:45 we begun math watching lecture. Then sorry after that I think I ate breakfast then I came back around 11:00 finished watching the entire lecture but then we're back. Let's do this. So question question one question question one by the way first I'm going to so they're ask given the following integral in a value of n approximate the following integral using the method indicated round your answer to six decimal places. So what we're going to do is open up desmos.

[00:21:14]This is going to be extremely vital for our calculations.

[00:21:18]Let's do this. So, oh wait, is there a way I can split the screen?

[00:21:41]Yes, there we go.

[00:21:54]Perfect. We're good.

[00:21:58]So, first trapezoidal rule. Let's open that. Where's the trapezoidal rule? It's right there. So, what's smart charging is on.

[00:22:11]Oh, don't show this again.

[00:22:14]Well, cool. I really appreciate it. So, let's write down our integral.

[00:22:20]We have the integral from 0 to 1. Where am I? Okay, good.

[00:22:29]0 to 1 of e to the^ 4x^2 dx.

[00:22:38]And we need to evaluate it at n= 4.

[00:22:43]So trapezoidal rule this is going to be this is going to be approximated too.

[00:22:51]Oh first let's calculate delta x. So what is delta x? delta x is going to be 1 - 0 / 4 1/4 or 0.25 depending on how we want to write our answer. So this is approximated to 0.25 25 / 2 times well oh and first what is what's going to be all right and we need n equals 4 so x sub0 will equal oh wait but hold on depends on no it actually does not depend on anything zero x sub1 equals = 0.25 x sub 2 = 0.5 x sub3 = 0.75 and x sub4 = 1 and we need n= 4. So uh yeah, this is probably going to take longer than we think. So this is going to be um f of x sub0.

[00:24:11]So this is going to be e to the power of um 4 e ^4 * 0^ 2 + e to the power of -4 * 0.25^ 25^ 2 + e ^ -4 * 0.5^ 2 + e to the^ -4 * 0.75^ 2 and actually I think this is wrong.

[00:24:52]Do we begin from from one or zero?

[00:25:03]No, it's zero. Oh, but you do it for Hold on, bro.

[00:25:13]So this is going to be oh and then do we do plus e to the^ -4 * 1^2 so simplifying okay in fact I'm just going to write this in desmos um 0 square is 04* 0 e to the okay so e to the That's 1 + e to the^ of 4 * 0.25^ squared.

[00:25:53]And in fact, here we're just going to do e.

[00:25:59]There you go. And here we're just going to do f of 0 plus f of 0.35 plus 5 + f of 0.75 plus f of 1.

[00:26:16]That looks correct. I want to see if I got the answer correct. Yeah, that's incorrect. So it looks like we did something wrong.

[00:26:33]What could we have done wrong?

[00:26:44]Uh oh. Six decimal places. So 1 2 3 4 5 six. So how how do we round from there?

[00:26:54]800 Um, oh, whoops. We forgot something very important.

[00:27:11]The twos.

[00:27:16]All right, that should be correct. So, I'm just going to input the Okay. All right, we have the right idea. Good. Good.

[00:27:28]Let's Oh, we But we we want the All right. One, two, three, four, five, six.

[00:27:37]If we do 10, do we still get it correct?

[00:27:39]Good. All right. Midpoint rule.

[00:27:45]Using the middle point rule, we get we get um 0.25 25 times in fact this is worth doing in Desmos 0.25 and we're going to have All right. So, first of all, we're going to have this 12 times 0 plus 0.25.

[00:28:38]That is going to be the middle point f of Oh, no, no, no, no. Is that only the x value?

[00:28:49]Hold on.

[00:28:51]Oh, that's right. That's only the x value. So f perfect.

[00:29:07]Plus f of this.

[00:29:16]Whoops. What? Why? Oh, right. And that's going to be 0.25.

[00:29:22]time 0.5 plus f of 0.5 plus 0.75 and this is going to be all the way until n. Oh wait, no. Pause pause. Uh that's one all the way to two I think honor standing. Okay, hold on.

[00:30:00]Oh, and then one more.

[00:30:11]Okay, good.

[00:30:16]Looks good. Unless I'm missing anything.

[00:30:18]I think that looks good.

[00:30:24]That's wrong.

[00:30:26]And why is that?

[00:30:49]Mhm.

[00:30:57]four right in between each of them. Uhhuh.

[00:31:09]Wait, what in the world is this? Plus K.

[00:31:21]Oh.

[00:31:54]Okay, this is one of those things that we got to debug now.

[00:33:45]Bro, Oh.

[00:34:10]Oh, that's why.

[00:34:14]Okay, there we go.

[00:34:28]Perfect. Now we just need Simpsons rule.

[00:34:54]In fact, I'm going to put right here.

[00:34:57]Since n is even, we can do this. So, we're going to get 0.25 over 3 times.

[00:35:17]Okay. So, f ofx sub0.

[00:35:21]Whoops. And what is x sub0? Is zero.

[00:35:23]Uh-huh.

[00:35:28]+ 4 uh times f of 0.25 I believe because yeah x sub 1 is 0.25 5 + 2 * f of 0.5 + 4 * f of 0.75 and then plus plus f of one.

[00:36:09]There we go.

[00:36:13]1 2 3 4 5 6.

[00:36:17]Very good.

[00:36:19]That was not complex at all.

[00:36:24]But I see that this is all about going quickly. It's all about going quickly, everyone. We can do this.

[00:36:33]What? Oh, it's that part. All right, let's keep going.

[00:36:43]A radar gun was used to record the speed of a runner during the first 5 seconds of a race.

[00:36:51]So t of s. So this is time based. Oh, I'm sorry. What's t distance? Distance based. Why do they have to use t?

[00:37:01]Distance based in seconds.

[00:37:06]So um what do we have? So t distance wait a radider gun was used to record the speed the speed of a runner during the first five seconds.

[00:37:23]So in 5 seconds so this is seconds right t of s seconds I guess.

[00:37:36]Anyways, use Simpsons rule to estimate the distance the runner covered during those 5 seconds. The distance. So, looks like we're going to have to integrate because here we have a speed, that's a velocity, I think. Uh, yeah, v of t. And to get the distance, we're going to have to integrate.

[00:38:01]Let's do it.

[00:38:07]Uh oh, cool. I see. I see.

[00:38:12]So, the way I see it, the way I see this question is, let's see. I'm confused. Why do they use a function to simply measure time?

[00:38:32]the distance the runner covered during those five seconds.

[00:38:38]Uh, okay.

[00:38:44]So, in 5 seconds he covered 10.9 feet.

[00:38:52]Sorry, I'm trying to understand this.

[00:39:40]Yeah. Mhm. 3 seconds. 35 seconds.

[00:39:58]Oh, I'm sorry. I'm dumb. The bottom the bottom row row is the the velocity, I think. All right, let's just go for it.

[00:40:11]We won't understand unless we begin.

[00:40:14]So, first, okay.

[00:40:18]All right, here we go.

[00:40:21]X sub0 equals Z.

[00:40:26]X sub1 let's say it equals 3.1 X sub 2 = 3.5 X sub3 = 6.95 X sub4 = 7.45 45.

[00:40:52]X sub 5 = 8.85.

[00:40:56]X sub6 = 9.75.

[00:41:03]X sub7 10.45.

[00:41:07]X sub8 = 10.55.

[00:41:13]X sub9 = 10.9.

[00:41:18]And finally x sub 10 = 10.9 two rule destinate the distance. Gotcha. So So these are our x's.

[00:41:36]These are our x's. Okay.

[00:41:40]So what do we see here? 1 2 3 4 5 6 7 8 9 10 11.

[00:41:49]Hold on. If I have uh a partition, one partition, two partitions, three partitions. If I have four points, that's three partitions. That's not four, right?

[00:42:02]We have three, excuse me, we have three intervals here. n= 10 and um delta x equals well this is going to be hold on so bro think this goes from time zero all the way to time five so I believe this is 5 - 0 over 10 or 12 cuz 5 over 10 is 12 1/2.

[00:42:43]I see. So I think that's all we need. So now what do we do? Use Simpsons rule and let's go. We have it right here. So delta x over 3.

[00:42:56]Okay. Yeah, I don't think we need to write that. Let's just write it. So we have 12 over 3 times f of 0 plus 4 * f of 3.1 plus 2 * f of 3.5 plus 4 * f of 6.95 plus 2 * f of 7.45.

[00:43:50]Okay, not bad. It doesn't seem bad yet.

[00:43:54]plus four times f of 8.85 plus 2 * f of 9.75 plus 4 * f of 10.45 plus 2 * f of 10.55 plus four times f of 10.9 plus f of 10.9. We did not multiply that last one because you do not multiply.

[00:44:41]Oh, sorry. No, no, no, no. That does not matter. Anyways, now we just have to let her rip in dasmos.

[00:44:52]So 1 over two all of that over three that's the same thing as 1 over six. So we're going to write that algebra proficiency proficiency is important. So we're going to take how can you Okay, let's say this is g of x.

[00:45:11]Fine. Fine. Let's see. That's f of x.

[00:45:14]Now we have g of x. What does it equal?

[00:45:19]Oh wait, pause. Pull the brakes. We don't even have a function.

[00:45:28]Why did it put it in F?

[00:45:31]Wait, am I stupid? Okay, hold on. Hold on. Pause. We went over six. In fact, I don't know why we wrote all of that. We just wasted a bunch. No, no, no. It's fine. we we fulfilled our our uh conceptual understanding but it was not necessary at all. So zero plus oh no no here in fact I'll just follow it over here. So, we have four.

[00:46:01]Wait, no, no, no, no, no. Wait, pause, pause, pause.

[00:46:05]Oh, okay. Wait. I think I did this wrong. X sub0 is 0. X sub Okay, so yeah, I did this all wrong. I understand why I got it wrong, so I'm not going to fix it.

[00:46:16]But it was wrong.

[00:46:19]4 * 3.1.

[00:46:24]So we have 0 * 0.

[00:46:28]Yes. Plus sorry we had 1 * 0 I believe.

[00:46:36]Plus 2 * 3.5 plus 4 * 6.95 dude. 6.95 + 2 * 7.45 + 4 * 8.85 + 2 * 9.75 + 4 * 10.45.

[00:47:08]Excuse me, that's rude. * 2 * 10.55 + 4 * 10.9 + 10.9.

[00:47:23]Now that my friends is our answer and that was red in the first try. Let's go. Do you all see we're are we are literally cooking in here. Let's go.

[00:47:36]Let's go. That and we go 40 what the what? 47 minutes.

[00:47:42]How is it even possible? There's Okay, that's crazy. You know, I remember when I was a kid and every time I played a Brawl Starters, my favorite video game at the time, I always complain time passes so fast when I'm playing video games, and time passes so slow when I'm doing homework. That never happens now.

[00:48:02]I see that when you're in college. In fact, time passes even faster when when you're doing homework, when you're coding.

[00:48:12]Bro, that's incredible. We got some physics. Now we can look smart. Let's go.

[00:48:20]So, we only go like 10% of the entire thing. Okay. Gotcha. Gotcha.

[00:48:30]Looks like a very nice day outside. I'm not going to lie.

[00:48:34]Look at the plugs.

[00:48:37]A very nice day.

[00:48:41]It's actually quite nice in this room.

[00:48:43]I'm not going to lie.

[00:48:53]Um, okay.

[00:48:58]Come on. How long we go? 48 minutes.

[00:49:24]Cool.

[00:49:29]All right, let's keep going.

[00:49:32]Consider the integral approximation t of t sub20 of the Okay, sorry. I'm just going to go to the bathroom real quickly and then we will lock in again.

[00:50:35]Here we go.

[00:50:54]So consider the integral approximation t sub20 of the integral from 0 to 4 of 4 e to the power ofx over 4 dx. Does t sub20 over overestimate or underestimate the exact value?

[00:51:13]t sub20.

[00:51:18]What is t sub20? Oh, the integral approximation. Why is it called like that? What is t sub9 or t sub 21? Okay.

[00:51:28]Find the error bound for t sub20 without calculating t subn using the result that error of t subn is less or equal then m * b - a cubed over 12 uppercase n^2 where m is the least upper bound for all absolute values of the second derivative of the function 4 e to the^ ofx4 on the interval on the closed interval a comma b. Wow. What am I reading? I don't know.

[00:52:06]Let's get some help to understand.

[00:52:13]Oh, we're bringing back concavity.

[00:52:54]Okay, let's do this. So to decide whether t sub20 underestimates or overestimates the integral, think about the concavity of f ofx.

[00:53:09]Guess let's put it in desmos.

[00:53:28]Gotcha.

[00:53:53]H yeah never goes negative.

[00:53:57]It's always greater than zero.

[00:54:17]An upper bound. Okay.

[00:54:45]What is B minus A though? What is A and what is B? Where do we get 0, 7?

[00:55:00]All right.

[00:55:02]I might have to skip this problem because I don't exactly understand.

[00:55:06]No, but I want to get it over with. Uh so we don't so we don't have to come back to it.

[00:55:15]All right.

[00:56:27]Gotcha.

[00:56:36]Okay.

[00:56:40]I see. I see. Okay. So that means hm Okay. But what is an error bound though?

[00:57:10]Okay.

[00:57:13]So, think about the concavity. Wait, do do we get the same function?

[00:57:22]Oh, hold on. All right. So, I'm just going to write the function down.

[00:57:36]Okay.

[00:57:40]So to decide whether t sub20 underestimates or overestimates the integral, think about the concavity of the function. Oh well of that was fake.

[00:57:57]Don't know why I did that.

[00:58:00]Okay. So let's think about it.

[00:58:03]we have e to the^ of -x over 4.

[00:58:11]So this is 4 over e to the^ x over 4.

[00:58:20]So e is growing but it's decreasing actually because it's one over.

[00:58:32]It makes sense, right? Yeah. So that's the concavity. Infinity all the way to zero, right? And that's exactly what we see. Infinity all the way to zero.

[00:58:44]Calculate f prime prime.

[00:58:50]I don't know. But okay.

[00:58:55]So first what is um what is frime of x?

[00:59:01]That's going to be uh 4 * well e * sorry e to the powerx over4 times the derivative of the inner function x4 that's going to be * -4 I believe so so 4 * 1/4 that is simply negative 1 so we have we're left with negative e to the^ of -x over 4 and then taking the derivative of that we get well I'll just open square brackets e to ^x over 4 * the inner function it's going to be - 1/4 once again so then -1 * 1/4 is simply 1/4* e^ x 4 is that what you got that's exactly what I got and by seeing the function we know that it's greater than zero for all x.

[01:00:03]So t sub 20 overestimates the integral.

[01:00:08]Good to know the upper bound. Oh, wait.

[01:00:10]But that's what they're asking us. So it's not good to know. It's it's actually essential. Anyways, the upper bound of the integral uh oh on the error.

[01:00:26]Let's put let's put this over here because it makes this look cool.

[01:00:31]Actually, no, it doesn't kind of.

[01:00:34]So, we have the absolute value of t subn minus the integral from a to b of fx dx.

[01:00:47]Oh, so the upper bound on that error meter it's going to be well why not right k sub 2 * b - a ^ 3 over 12 n^2 where k^ 2 is an upper bound for double derivative of f ofx on the closed interval ab. An upper bound for uh the absolute value of the double derivative on 0 to 7 for us is going to be 0 to 4 is 0.25.

[01:01:39]I'm sorry, what are we doing? Are we are we replacing Oh, the upper. So, did he replace zero?

[01:01:56]Negative 0 before that's negative 0.

[01:01:58]That's 0. E^ 0 is 1. 1/4. Gotcha.

[01:02:05]Oh, I see.

[01:02:10]Well, I believe we also get 0.25.

[01:02:16]Wait and 0.25 squared that's going to be like I don't know oh sorry sub 2. So we have 0.25 * 4 cubed over 12. And what's our n? Are you going to tell us the n or not?

[01:02:35]Well, you got 20. But why 20? Where did the 20? Oh, that's right. We also got 20.

[01:02:45]Well, let's just type that in, right?

[01:02:48]Whoops.

[01:02:52]0.25 * uh 4 cubed or 64.

[01:02:59]That's 16, right?

[01:03:02]Time 400, we get this.

[01:03:08]Is that our answer? And that is our answer. Oh, but we did not answer this.

[01:03:14]Well, that overestimates.

[01:03:17]And we're correct because we're so good.

[01:03:26]Let's go. That's three questions out of 21.

[01:03:31]Oh boy.

[01:03:34]Let's do this. I mean, these aren't hard questions. Actually, I'm not having to use a lot of help. In fact, I don't think I had to use any of help except for the third question that I had no idea what they were talking about. But no, I mean, this is easy. This is actually not very bad.

[01:03:53]So, use six rectangles, meaning that n equ= 6. Whoops.

[01:04:06]uh to find an estimate of each type for the of each type for the area under the graph of f from 0 to 12.

[01:04:27]I see Uh h hold on. Use six rectangles to find the estimate of each type. What does it mean by each? Okay, it's fine. It's fine. So let's see. So this thing is decreasing.

[01:05:00]Are they going to tell us uh oh of we of each type? Does that mean like of each rule? No, I don't I don't think so.

[01:05:08]Hopefully not because that would be pretty long. All right. Take the sample points from the left end points left. Okay.

[01:05:21]So we get there you go. So we get um so first what is delta x 2.

[01:05:46]So we have 2 times times uh so what is x sub0?

[01:06:01]Oh yeah left end points right. So yeah, we begin with uh f of x sub0. f of x sub0 looks like it's zero.

[01:06:12]Plus f of x sub 1.

[01:06:19]So 2 + 4 + 6 8 + 10. 1 2 3 4 5 6. Mhm.

[01:06:27]That's good.

[01:06:32]really easy. Uh 10 20 30 2 * 30 is going to be 60.

[01:06:44]I think that's correct. Is that how they wanted to do it? Perfect. Okay, we're good. Let's keep going.

[01:06:52]Is your estimate an underestimate or overestimate of the true area? Well, it depends what the true area is, right?

[01:07:08]Let's think about it.

[01:07:12]Well, we have 1 2 3 4 5 6 7 8 9 10 11 12 24.

[01:07:18]We have 24.

[01:07:20]We have uh 36 35. We have 35.25.

[01:07:28]35.25 46. Oh, wow. Yeah, I know. I'm not sure.

[01:07:37]It's okay. I'm just gonna fight the No, no, no. Hold on. So, underestimate or overestimate? Well, bro, it depends on Let's see. Uh, I mean, do you want me to find the actual estimate?

[01:07:54]I think I'll need the actual function for that, though.

[01:08:10]underestimate or or overestimate of the true area.

[01:08:21]Let's see how they do this.

[01:08:39]Okay.

[01:08:42]All right. It's fine.

[01:08:48]The right end points is going to be 2 * 2 + 4 + 6 + 8 + 10 + 12.

[01:08:59]This going to equal 2 + 12 14 28 uh 42 2 * 42 84. So we get 84. commit points and get 6 3 + 10 5 uh 7 9 10 10 25 15 H this is overestimate and overestimate.

[01:10:09]Oh, this is also 84.

[01:10:24]So we have 2 * 14 + 14 that's 28 + 14 that's 42. 42 * 2 that's 84 isn't it?

[01:10:41]Looks like that's an underestimate. So I should have get What am I missing here?

[01:10:59]and the 50 is also incorrect.

[01:11:03]Let's see what what did we do wrong?

[01:11:09]Oh, wait. Whoops.

[01:11:14]uh and this was + 12 10 + 12. So therefore 22 11. There we go.

[01:11:28]So 12 + 12 24 + 12 36 36 * 2 is 72.

[01:11:40]And why? Let's see, bro. What in the world is going on?

[01:11:51]Let's see. Let's see. Think. Come on, T.

[01:11:54]Think.

[01:11:56]Yeah. So, delta x is two.

[01:12:03]Let's see what am I missing. So, we have Yeah. 84. Am I doing something? No.

[01:12:17]Okay. Well, obviously I'm doing something wrong.

[01:12:46]Oh, wait. What?

[01:12:50]Uh, wait. Oh, great. I mean, these are not X values I'm adding. I'm adding y values.

[01:13:06]What are the y values here?

[01:13:22]Oh, to estimate.

[01:13:29]So we begin here that is 1 2 3 4 5 6 plus that looks like a 5.5.

[01:13:51]That looks like a five.

[01:13:55]That looks like a 4.25.

[01:14:02]That looks like a 3.25.

[01:14:08]And that looks like a two.

[01:14:11]So adding this up.

[01:14:22]52.

[01:14:28]And what? Oh, now it's only mid point.

[01:14:31]So it looks like we are also going to estimate the middle points.

[01:14:36]Yeah. So this is actually wrong. Don't know why we did that.

[01:14:43]Mid point. So it's going to be here.

[01:14:49]It's going to be six.

[01:14:56]Okay. 6 plus 5.75 5.25 4 4.75 3.75 2.75 56.5.

[01:16:02]There we go. Very good. Very good.

[01:16:06]That's much better.

[01:16:08]I started streaming 67 minutes ago.

[01:16:11]Wait, 67. Oh, what are the chances?

[01:16:15]Oh, well, thanks for letting me know.

[01:16:18]Thank you. Yeah, 67 has been haunting me. My iPad yesterday was 67%.

[01:16:25]My brother's uh computer was 67%, too.

[01:16:29]What else was 67%. Oh, I think I got a 67 result in in my coding project uh in in a program I did. Oh, and also my reading was 67%. I finished it yesterday at 12 a.m., but we got a I mean, we got an 100, but before that we had a 67%.

[01:16:54]So, incredible, isn't it? 67 is everywhere. I wondered what Bible verses of 67 in it.

[01:17:04]Crazy. Crazy, isn't it? How has 67 or 67 haunted y'all?

[01:17:13]All right. Well, that's fun.

[01:17:19]Okay.

[01:17:21]Okay. Okay. Well, I believe we should get back at it because even though we're cooking, we aren't close to done.

[01:17:30]Let's do this. Shouldn't be too hard.

[01:17:34]Um take the sample points from the left end point.

[01:18:04]So we have oh and by the way from one to nine so delta x okay hold on uh and four rectangles n = 4 from 9 to uh 9 - one all of that over four two right I'm just going to get rid of these things there we go two delta max is two and we have four rectangles. So left end points we're going to get and in fact maybe it's better split the screen.

[01:18:49]Um there you go.

[01:18:53]So we have two times.

[01:19:01]So left end point we begin at so this is f of one and that's two plus 2.25 plus 2.75 plus 3.5 plus 4.25 25.

[01:19:32]Oh, we whoops without four. There we go.

[01:19:35]So, this is going to be 21. And let's see whether it is an underestimation or overestimation. So, two times we get 2.25 + 2.75 + 2.5 + 4.25.

[01:19:56]This is 25.5.

[01:20:00]using the trapezoidal rule.

[01:20:03]Let me see if I remember. So, delta x is two 2 over 3 times times what?

[01:20:15]Uh oh boy, what was it?

[01:20:23]4242. No, that was the Simpsons rule.

[01:20:27]This was Oh, okay. I remember. So we we we begin at two I believe plus 2 * 2.25 + 2 * 3.5 + 2 * 4.25.

[01:20:48]I believe that's correct.

[01:20:51]Oh, whoops.

[01:20:55]No. going.

[01:21:12]There we go.

[01:21:26]Delta X is two.

[01:21:36]You know, first of all, is this correct?

[01:21:41]So it's only the trapezoidal rule. Uh I thought I ended the live stream.

[01:21:58]So let's remember oh it's over two. So here we have nothing there.

[01:22:17]27.5.

[01:22:21]So under under Oh, hold on. We So the T of four was or or or the T of 20 was a trapezoidal. Oh, I didn't understand that.

[01:22:48]Oh, and the 27.5 is incorrect.

[01:22:52]So this overestimation then. Okay.

[01:23:07]Oh, and we begin.

[01:23:11]Oh, right, dude. There we go.

[01:23:18]That looks correct.

[01:23:23]Very good.

[01:23:25]Very good. We continue.

[01:23:32]So, we have 15 16 questions left.

[01:23:37]Okay.

[01:23:39]I mean, this isn't too bad, but maybe pulling a game is not too bad because playing a game is kind of fun.

[01:24:07]Oh, I thought I could kill him. would have been pretty cool.

[01:24:12]Oh, a free food for me. Ah, okay. Well, still.

[01:24:20]So, first place is 25,000. Oh.

[01:24:39]Oh, dude. Look. I'm getting all this food.

[01:24:50]Dude, I feel I feel so bad for him.

[01:24:56]No. What? How? That's not fair, bro.

[01:25:01]That's so not fair.

[01:25:04]Okay. Oh, that was fun.

[01:26:27]bro. What?

[01:26:40]Oh, yes. How much food?

[01:27:54]bro. What?

[01:28:40]Dude, that's not cool.

[01:29:36]Let's be pink this time. We are distinct, guys.

[01:29:40]Oh, let's go.

[01:30:09]dude. Why do you have to do that? Don't do that.

[01:30:56]What?

[01:31:52]Let's go.

[01:33:11]Oh.

[01:33:16]Oh, dude. Why did this guy have to be here?

[01:34:14]Why did he do that?

[01:35:15]Bro, don't stop.

[01:35:19]Why am I complaining so much, bro?

[01:37:34]Let's go.

[01:38:05]bro.

[01:38:07]All right, guess we can play one more game.

[01:39:10]Let's go.

[01:42:53]H.

[01:42:57]Okay.

[01:43:27]Ah, great. All right. Well, that was fun, everyone.

[01:43:32]That was pretty fun. But we are back. We are back to more calculus. Why is bro streaming? Well, because it helps me focus with my homework. Four likes. A thank you, guys.

[01:43:50]twin. All right, let's go. Let's go.

[01:43:52]Yes, this is a study stream.

[01:43:55]This is a study stream. We were taking a break, but but now we're back and we're going to lock in.

[01:44:02]So, here we go.

[01:44:06]The graph of a function f is given below. Estimate Oops.

[01:44:13]Estimate the integral from 0 to 8 of f ofx dx using four sub intervals with uh right right end points, left end points and midpoints. Very easy. Okay. So I think we're simply supposed to uh observe the graph.

[01:44:40]So a delta x is going to be 8 - 0 over 4 and that's two.

[01:44:47]Let's see. So let's estimate this using left end points. Let's see. Desmos is right here. This graphing calculator. So since we have two so the left end points is going to be uh first we're going to have f of zero which is going to be two plus then we're going to be go down here.

[01:45:08]That's gonna be one plus two right here. That's two plus. Wait, this is left end points.

[01:45:19]Left end points. Hold on. Right. Mhm.

[01:45:23]Plus two. We're going to go down here and that's going to be -2. And right, it's assigned area so that's okay. And then two, we're going to go to one plus one. So I believe this is going to be eight.

[01:45:38]Using right end points we get uh 2 + 1 oh wait six right here 2 + 1 + 2 - 2 + 1 we actually begin at this one right here and we here we get four and the middle point rule well quite easy right and we're evaluating y values f ofx values Um, 12 0. Oh, whoops. 12 2 + 1 + 12 1 + 2 I believe plus 12 wait one two - 2 + 12 -2 + 1 and we get five. That should be correct.

[01:46:46]And they're all wrong. That's crazy.

[01:46:51]And how did this happen?

[01:46:54]Oh, hold on. Wait. So, this is right end point. So, four. This is six.

[01:47:02]Well, that should be correct. These were correct, but then the midpoints are incorrect.

[01:47:09]the midpoints. Uh, yeah. Delta x 12 + 1 plus 12 1 + 2 + 12 2 - 2 + 12 - 2 and 1. And that's five.

[01:47:32]And what could be wrong over here?

[01:47:37]middle point. Oh. Uh, well, we could also do two plus Whoops. Actually three + 2 = 2 + 1 2 1 minus one. That gives 10.

[01:48:17]Is that for real?

[01:48:21]Hold on.

[01:48:25]So, what was wrong with with what we did?

[01:48:31]The minute point rule.

[01:48:34]Yeah, we did 2 * um 2 + 1 * 12 of that.

[01:48:47]Interesting. Okay. Well, let's continue. Given the following graph of the function y = f ofx and n= 6.

[01:48:58]Answer the following questions about the area under the curve from x= 0 to x= uh six. Okay, so we observe it goes high then lower then a little bit higher. Round your answer to within two decimal places if necessary but do not round until your final computation. I see. So use the trapezoidal rule uh to estimate the area. So we get so first of all what is delta x delta x is going to be 6 minus 0 over six so that's one need six rectangles. Gotcha. So the the delta x the width is going to be one.

[01:49:48]Beautiful. So then we will do we will do now all right what's next?

[01:50:00]So what's f of zero? Wait do we do from f of zero? Oh yeah all the way to f of n. Uhhuh. So so what is f of zero? So 2 plus 2 * uh 3 plus 2 * 4 + 2 * 3 + 2 * 2.5 Whoops.

[01:50:32]+ 2 * 2 + 4 believe that's 35. Hold on. Wait, I might be wrong. So 2 * So 2 + 2 * 3 right there.

[01:50:51]Am I doing it correctly? I think so.

[01:50:54]Plus 2 * 4 + 2 * 3 + 2 * 2 5 + 2 * 2 + 4.

[01:51:11]Use Simpsons rule.

[01:51:14]So here we're going to have Oh, hold on. Wait, no, no, no. Um, here we have delta x over two. So we're going to have 1/2. There we go. So 17.5.

[01:51:30]Here we're going to have over three, right? Delta x over 3. So 1/3 times. So, we're going to have 2 + 4 * 3 + 2 * 4 uh 4 2 4 + 4 * 3 + 2 * 2.5 + 4 * 2 + 4.

[01:52:04]So, here we get 17.

[01:52:09]Oh, boy. Wow. What happened here?

[01:52:16]Oh, wait. Oh, so we have 1/3.

[01:52:31]Um, then we have two 42424.

[01:52:46]Was it wrong to estimate the area?

[01:52:59]We have one/3 two Oh, great.

[01:53:24]And here we had one half bro.

[01:53:56]I'm not sure. Oh, round to two decimal places. Do not round until your final computation.

[01:54:13]I don't think I have to run.

[01:54:23]Yeah, that's exactly what we did. Wait, but why did you get this number?

[01:54:53]Bro.

[01:55:32]I don't see why this is wrong.

[01:55:57]just like as like that.

[01:56:00]What did we mix them up?

[01:56:16]It's the trapezoid rule to estimate the area.

[01:56:21]So we had one and then you divided the the the delta x and f of x sub0.

[01:56:29]That's f of 0 or one.

[01:56:36]Oh, sorry. Or two.

[01:56:43]I literally have no idea why this is wrong.

[01:57:33]Let's see how they did this.

[01:57:36]Um, am I not sharing my screen?

[01:57:48]Oh, I thought I was sharing my screen. I don't know what happened. Oh, I think what happened is I I accidentally I turned off my computer. My bad.

[01:58:38]All right. So everything is correct except so let's see one sorry two that's two 3 4 3 that's 2.5 that's two but that's four dude Guess I'll have to ask AI or something, but I have no idea why this is wrong.

[01:59:18]Why in the world are my answers wrong?

[01:59:26]Verifying that I'm human. Do I look like human, guys?

[01:59:31]All right, I'll admit. I guess I do act like a robot sometimes.

[01:59:43]ever goes above three.

[01:59:47]Well, it does clearly.

[01:59:51]Wait, what is that? Five.

[01:59:55]Wait.

[01:59:58]Okay.

[02:00:04]Three.

[02:00:06]five.

[02:00:10]Then we're back to three and 25. Oh, so this is 18.5.

[02:00:25]Round to two decimal places.

[02:00:29]I don't know what happened. I was like hallucinating or something. But I swear I saw those at 4.

[02:00:37]What in the world was that? That's weird.

[02:00:42]Okay, I got to go quick because these questions are really easy. I don't know why I'm taking so long.

[02:00:50]Uh, wait. And and how long do we go? We go two hours. Okay.

[02:00:54]So, estimate the area under the graph in the figure by using the trapezoidal rule. the midpoint rule and the Simpsons rule each with n equals 4.

[02:01:06]So n equals 4.

[02:01:09]Um gotcha. In fact, let's do it in the iPad. I I believe it's better here.

[02:01:20]So uh let's see.

[02:01:23]So we have n= 4.

[02:01:28]Okay. So first the trapezoidal rule we have okay and delta x equals 4 - 0 over 4 and that's equal to 1 delta x = 1 this minus here. Yes. So wait and what is next? Oh yeah, I just went blank. So we have 12 in fact I'll just write this in Desmos. We have 12 times and then we have f of x sub0. f of x sub0 * f of x sub0. f ofx sub0 is zero plus 2 * oh and right one plus two * whoops and this is not one don't know why I put that this is going to be three times that's sorry plus 2 * 5 plus 2 * 3 + 1 11.5.

[02:03:05]The middle point rule, we simply have delta x. Delta x is one. So, we're not going to multiply anything.

[02:03:40]Okay.

[02:03:42]So, midpoint rule we have 12 * 0 + 3 + 12 * 3 + 5 + 12 * 5 + 3 + 12 * 3 + 1 11.5 as well.

[02:04:13]Then here we have in Simpsons rule we're going to have 1 over 3.

[02:04:21]We're gonna have 0 + 4 * 3 + 3 * 5 + 4 * 3 + 1.

[02:04:35]All right, this getting freakishly easy because the last questions, the last assignments were incredibly difficult, but this is tough.

[02:04:47]Thank you.

[02:04:51]Okay. Thanks a lot whoever you are.

[02:04:57]Thanks guys.

[02:05:00]See we go two hours.

[02:05:05]Uh the left right trapezoidal and midpoint rule approximations were used to estimate the integral from 0 to 2 of f ofx dx where f is the function is the function whose graph is is shown below. Gotcha.

[02:05:27]Okay. The estimates were 0 point uh a 7811 0.8675 0.8632 and 0.9540.

[02:05:41]And the same number of sub intervals were used in each case.

[02:05:46]The same number of sub intervals were used in each case. Same number of sub Oh, right. Yeah. Which will produce which estimate? I see. Okay.

[02:05:58]Um, oh gosh. Okay, so let's see. So, oh yeah, we we had a rule decrease by a factor of K.

[02:06:15]So, the trapezoidal should be the most accurate one. Let's see. Uh zero to 0 to two.

[02:06:49]Okay. So these should be almost the same. So, I'll put this and that All of them are wrong. Yo, am okay. Let's see.

[02:07:54]So if we do left left and if we do right.

[02:08:02]Oh, so the left hand should be like the greatest and the right hand should be the least. I believe midpoint should be somewhere in between.

[02:08:20]Is this it?

[02:08:22]All right. So, I was right about the the greatest and the least.

[02:08:27]It looks like the trapezoidal rule is going to be this 75.

[02:08:33]I mean, how is I supposed to know that?

[02:08:36]Oh, hold on. The chop is okay.

[02:08:40]Okay. Between which two approximations does the value lie?

[02:08:49]Well, it must lie between the middle point and the trapezoidal.

[02:08:55]There you go. That seems correct. Yeah.

[02:08:58]Oh, very good. Consider the four functions shown below. On the first two, an approximation for the integral from a to bfx dx is shown.

[02:09:08]For graph number one, which integration method is shown? This seems like it is uh left.

[02:09:19]Let me just make sure.

[02:09:24]Yeah, it's always hitting the the left corners.

[02:09:29]Is this method an over or underestimate?

[02:09:32]This is going to be I believe it's going to I believe it's going to be an over an overestimate clearly since our function is decreasing or it's decreasingly increasing. Okay. Well, it's decreasing its rate of change.

[02:09:53]So the the rectangles are going to be bigger.

[02:10:00]For graph number two, it's going to be this is midpoint over or underestimate. Well, this is um Oh, I I see.

[02:10:18]Looks pretty accurate, actually. Let's see. Uh here it's missing this part.

[02:10:32]I believe it's under on the copy of graph number three. Sketch an estimate.

[02:10:40]Sketch an estimate with n equals two subdivisions using the right rule. So we are going to get a rectangle here and the corner is going to touch the graph. Then we're going to have another rectangle aligned perfectly where the right corner touches the graph. That's how it's going to be if we were to do that. Is this method? Well, this is going to be um H.

[02:11:08]Oh, wait. Hold on. Wait, wait. Is this under the graph?

[02:11:15]Oh, well, if that's the case, then then we're going to have it's going to go under here.

[02:11:33]It's a little hard to tell.

[02:11:36]Oh, okay. Well, using this same logic over here, Yeah. Well, that's an overestimate right here. We're going to have we're also going to have an overestimate sketch with N to using the trapezoidal rule.

[02:12:03]Trapezoidal. Uh well I I don't think it's going to be any right.

[02:12:12]Hold on. In the last problem trapezoidal was bigger than the middle point.

[02:12:20]May maybe it's going to be over.

[02:12:29]So it's under. Okay. That's under and under like that.

[02:12:58]Hm. I thought it was going to be under over actually.

[02:13:03]Oh, okay. Hold on.

[02:13:07]And I don't know. And this Oh, using the trapezoidal.

[02:13:17]I mean, isn't it perfect? Unless I'm missing anything. Oh, well. Okay, I see.

[02:13:22]Uh, well, that looks like it's going to be okay. I wonder why that's under.

[02:13:32]It's okay. We got to keep going now.

[02:13:37]Uh, okay. How many questions do we have left? We have 10 11. We have 11 questions left. All right. Fine. Let's lock in. Estimate the integral from 0 to 1 of cossine x^2 dx.

[02:13:52]Okay.

[02:14:03]So from 0 to 1 of cossine x^2 dx using first the trapezoidal and then the midpoint rule and then and we're going to have n = 4 So and we have okay three perfect.

[02:14:36]So first the trapezoidal rule. So uh delta x oops not dx delta x.

[02:14:47]It's going to be 1 - 0 over 4 = 1/4.

[02:14:52]And then so then we're going to have 1/4 over two.

[02:15:04]Wait 1/4 over two or multiply by 1/2.

[02:15:07]Uhhuh.

[02:15:09]Times so this is going to be cosine of of zero. Oh, hold on. trapezoidal rule.

[02:15:23]So f of x sub0. So so that is cossine of 0^ 2 + 2 * cossine of 1. So 12^ 2.

[02:15:45]Okay. Oh, sorry. So first x sub0 is going to be zero. X sub one is going to be 0.5.

[02:15:52]X sub 2 is going to be 1. X sub3 is going to be 1.5 and X sub4 is going to be two plus cosine uh Whoops.

[02:16:17]+ 2 cossine of of 1^2 plus 2 cosine of 1.5^2 plus cossine of 2.

[02:16:41]Putting this in desmos, let's see what what we get. It goes down to zero plus two.

[02:17:01]Okay. Will we get that?

[02:17:04]Uh yeah, I think that's correct. Right.

[02:17:06]Mhm. So, five decimal places. One, two, three, four, five.

[02:17:17]And then we do the middle point rule.

[02:17:20]So we're going to have one, right? Because that's delta tax.

[02:17:29]Uhhuh. All right. In fact, here I'll just write it down over here. We have 12 0 + 0.5 plus 12 0.5 + 1 + 12 1 + 1.5 + 12 1.5 + 2 4 What in the world?

[02:18:20]Wait. And this is until one. Wait.

[02:18:23]Whoops.

[02:18:25]You think? Okay. I think I did this all wrong.

[02:18:30]So, first we're going to have 25 here, 0.5 there, 0.75 there, and one. There we go. Okay. Looks better. Then here we're going to Oh, that's not better actually.

[02:19:05]Well, I'm just going to copy this.

[02:19:09]One, two, three, four, five.

[02:19:12]So we do this three and is correct. Both of them are incorrect.

[02:19:27]N= 4.

[02:19:36]Oops, I forgot to square this.

[02:19:53]That's correct. There you go. And why isn't this correct?

[02:19:57]Because Oh, hold on. Whoops.

[02:20:04]Cosine of this Oh, there we go.

[02:21:12]Hm.

[02:21:13]Why?

[02:21:34]Let me see if I did everything correctly.

[02:21:58]H I don't know what's going on.

[02:22:17]>> Yeah.

[02:22:18]Let's see. H.

[02:22:44]What happened here?

[02:22:49]Oh, 0.25 that is Oh, yeah. You add the x values and then you you evaluate. Uh-huh.

[02:23:28]And do we have to to buy to know what in the world is going on here, bro?

[02:23:54]Let's see. Let's see what could be wrong.

[02:24:16]So, this is wrong, right? Yeah.

[02:24:19]Let's ask why this wrong. Shouldn't stay here.

[02:24:58]Wait. Whoa. Oh, 14th.

[02:25:01]It's 1/4. It's not one.

[02:25:20]All right, that should be correct. There you go.

[02:25:26]You forgot. Did you get your X hats for middle points?

[02:25:33]Oh, actually, yeah. I think I I forgot the delta x.

[02:25:39]Um I think I multiplied by one rather than 1/4. Yeah, like learn. Well, I haven't learned it yet. So that's why I can't use it. I wonder how you'd use it here.

[02:25:57]Interesting.

[02:26:58]Oh, seven out of five parts haven't been answered correctly.

[02:27:26]04.

[02:27:28]Oh, so it's greater.

[02:28:05]That's what Easter Wait, if I put submit answer Oh. Oh, right. There's overestimate as we just saw. Uh-huh.

[02:28:32]This Oh, wait. No, no, no. It's uh let's see. So since T4's underestimate must be less.

[02:28:42]Well, there we go.

[02:28:45]Actually, there you go.

[02:28:52]Yeah. Well, I'm going to go to the bathroom and then we will continue to finish this.

[02:30:34]Let's see. Is it possible? So whoops.

[02:31:00]Probably Hold on.

[02:31:36]Right. Mhm.

[02:31:43]Yeah. I guess so. It it is the average Oh, so it's not possible that they're overestimates.

[02:32:08]Now, both of them are overestimates.

[02:32:10]Oh, let's see. And why is that?

[02:32:15]I'm not sure. I guess I'll have to think about that. The absolute value of the error using trapezoidal rule for approximate integral it's about twice. So it is true trapezoidal rule for approximating a definite integral interesting. Okay.

[02:32:39]And then here. So, okay. It looks like consider the approximation for the in integral of sorry the integral from a to b of f ofx dx where f ofx is negative decreasing and concave up in the interval in the closed interval a to b.

[02:33:07]So f ofx is negative. Okay. Decreasing and concave up. Gotcha. Okay. Enter O for each statement if the approxim approximation is an overestimate and U if it's an underestimate. Write reman sum.

[02:33:29]Right. Uh well that's underestimate.

[02:33:34]Left is over. Trapezoidal over or under? Under. I think midpoint.

[02:33:51]Oh, concave up. So, it's going to be under then as well.

[02:34:01]Is a trapezoid a rule wrong? Probably.

[02:34:05]There you go. That's what I thought.

[02:34:08]Consider the approximation for the integral from a to b of f ofx dx where f ofx is negative. Oh, increasing and concave down. Oh, so it's the left half.

[02:34:21]It's the left half and the interval in the closed interval a to b. Oh, overestimation and u underestimation.

[02:34:28]So negative, increasing and concave down. There's that midpoint rule.

[02:34:35]Midpoint rule. Um same thing think.

[02:34:46]It's probably going to be underestimation. Right, it's going to be overestimation.

[02:34:56]left is going to be underestimation.

[02:34:59]Trapezoidal is going to be underestimation.

[02:35:04]Says trapezoidal overestimation this time. Gosh.

[02:35:09]Oh, so it is under. Oh, this is the midpoint rule overestimation.

[02:35:17]And let's think why that is.

[02:35:22]It's over. It's an overestimation.

[02:35:32]Okay, I think I can see it. All right, another one of these. So f ofx is positive. All right, so now it's above the x-axis. It is increasing and concave up. So the second half, not the first half, but the second half in the closed interval.

[02:35:48]All right, left remnum.

[02:35:58]That's going to be underestimation. The right is going to be overestimation. The midpoint is going to be overestimation. And the trapezoidal is under this time.

[02:36:20]All right. So maybe it's over then, right?

[02:36:23]Oh, so it was under. So is the midpoint under then?

[02:36:29]Oh, for real?

[02:36:32]So is the left and right wrong?

[02:36:41]I mean it's positive. It's increasing and concave up. Concave up.

[02:36:49]So increasing the left reman sum. I mean if we have something like that we're going to have something like that like that. So that's clearly an underestimation if we have right That's clearly an overestimation. If we have a midpoint, it's probably going to be the overestimation as well. And trapezoidal, we don't want one more.

[02:37:55]Mhm.

[02:38:10]Midpoint gives an underestimation.

[02:38:21]Oh.

[02:38:26]So, so this under and this is over.

[02:38:34]I see. Okay.

[02:38:39]Now, decreasing. So if we have a decreasing so in the first half if we have this left remman we're going to have that.

[02:38:56]Oops. All right. It's fine. Well that's going to be an overestimation.

[02:39:02]Well the right is therefore going to be an underestimation.

[02:39:06]The middle point it's going to be which is going to be bigger. Um, I think that's going to be over and this is under and Oh, there you go. Mhm.

[02:39:42]There we go.

[02:39:44]Now positive and increasing and concave down.

[02:39:53]So positive increasing left that's going to be that is going to be an underestimation right is going to be overestimation. The midpoint is going to be. So if we have under and over like that. Okay.

[02:40:36]positive decreasing and concave down. Oh, okay.

[02:40:44]So, the the left is going to be over.

[02:40:56]Right. Is going to be under, I think.

[02:40:58]Right.

[02:41:00]Right. the middle point rule under and so it's going to be over I believe sign so it's going to be over and it's going to be under then there we go negative so now it's under increasing and concave up okay so left end point oh wait negative increasing and concave up.

[02:41:33]So, this is going to be underestimation overestimate. Oh, wait. Hold on. Wait.

[02:41:39]Pause. Pause. But no, actually, no. It's not going to look like that. It's going to look like this. So, left.

[02:41:48]Oops.

[02:42:09]like that. So that's going to be an overestimation. Actually, that's going to be under what is it going to be like that?

[02:42:37]Yeah, that trapezoid and midpoint so annoying. Uh let's see an underestimation.

[02:42:51]Oh, right. Because it's in the second half. Okay. No, it it makes sense. So, this under in trapezoidal think I think that's it. All right. Fine. So, that's also under then.

[02:43:06]Excuse me, bro. So if we do that gives an overestimation.

[02:43:17]Yeah. About the right. That's going to be wait right is over. Why?

[02:43:30]That's clearly under.

[02:43:35]I'm confused.

[02:43:38]Am I imagining it in incorrectly decreasing? So it's negative decreasing.

[02:43:48]Okay. Left reman sum that's going to be under over.

[02:43:59]Trapezoidal it's going to be over.

[02:44:02]That's under.

[02:44:04]Great.

[02:44:12]lie above the curve. Uh-huh.

[02:44:28]Overestimation.

[02:44:32]Submit point.

[02:44:36]under then.

[02:44:39]Oh, and I forgot the left and right are opposites.

[02:44:44]Okay.

[02:44:47]Biochemical engineers are designing a new athletic shoe to analyze the impact absorption. They record the vertical ground reaction force F of T1.

[02:45:07]f oft exerted exerted by a runner's foot on a forest plate. The force in Newtons is measured every 0.1 seconds.

[02:45:18]Okay. The data for one full step is recorded below. Okay. So, x sub0 is going to be zero. X sub1 is going to be 0.1. X sub2 is going to be 0.2. X sub3 is gonna be 0.3. Okay. And on right um use the trapezoidal rule. Okay. So and this going to be from zero all the way to 0.6. Right? All right. So n equals 6.

[02:45:49]Um, so here we're going to have 0 delta x = 0.6 - 0 over 6 and that equals 0.6 / 6 or 0.1, right? Mhm.

[02:46:12]Uh 0.1. So trapezoidal we're going to have um 0.1 /2 times.

[02:46:24]So f of x sub0 it's going to be 0 + 2 * 250 plus 2 * 380. All right. And then on right 0.1 over two 0 + 2 * 50 + 2 * 390 + 2 * 240 + 2 * 470 = 2 * 110 + 0 146 impulse. Oh to estimate the total impulse generated during this step.

[02:47:08]Impulse is defined as the integral of force with respect to time. J equals All right. Well, I think that's it.

[02:47:17]All right, cool.

[02:47:20]That actually was much shorter than I thought.

[02:47:26]All right, awesome assignments.

[02:47:31]And that's actually a very easy one.

[02:47:35]Okay, cool. So, let's go here. Let's refresh.

[02:47:41]Let's go to grades. Let's collect our award.

[02:47:46]26 out of 26.

[02:47:49]Very good. Very cool.

[02:47:58]Awesome.

[02:48:00]Well, that was not bad.

[02:48:23]Okay. So tomorrow we have improper integrals and yeah I think that's it. Let's see.

[02:48:33]Is improper integrals hard?

[02:48:36]Let's just look at the PDF.

[02:48:42]Okay. It's just a convergence. Yeah. So I think I'll have to work on this trigonometric integrals and substitution and partial fractions. These ones I think I I already have. Well, but yeah.

[02:48:52]and the test this Thursday. We have to practice. Well, everyone, I want to thank you all for watching this video, this live stream with me. It was a lot of fun being with y'all. Uh, and I will see you all next time. Okay, I'll see youall next time.