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(a) Find the approximations $ T_{10} $ and $ M_{10} $ for $ \displaystyle \int_1^2 e^{\frac{1}{x}}\ dx $.(b) Estimate the errors in the approximations of part (a).(c) How large do we have to choose $ n $ so that the approximations $ T_n $ and $ M_n $ to the integral in part (a) are accurate to within 0.0001?

a. $T_{10}=2.021976$$M_{10}=2.019102$b. $\left|E_{T}\right| \leq .0068$$\left|E_{M}\right| \leq .0034$ c. Trapezoidal rule: $n=83, \quad$ Midpoint rule: $n=59$

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 7

Approximate Integration

Integration Techniques

Campbell University

University of Nottingham

Boston College

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Okay, I start. I want to find the t's up. 10. I'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'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's a function here, here to function. So we want to know. Okay, were Kay's. So we need to know the upper ball off the second directive. Right? Let'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's because the directive, right? You prime We plus you re price right and is to e Thanks. Plus right. Plus, there's a negative Negative, that'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'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'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're the three, right? And use a formula here. We can find a yam. It's just 1/2 off the e t. My biggest fear is that 24 years, so yeah, it's 3.4 times tend towards an active three. Here's party Parsi. How long do we have to choose in such that they'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's the formula hand just technically. And there'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's all.

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