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(a) Find the approximations $ T_[10} $, $ M_{10} $, and $ S_{10} $ for $ \displaystyle \int_0^\pi \sin x\ dx $ and the corresponding errors $ E_T $, $ E_M $, and $ E_S $.(b) Compare the actual errors in part (a) with the error estimates given by (3) and (4).(c) How large do we have to choose $ n $ so that the approximations $ T_n $, $ M_n $, and $ S_n $ to the integral in part (a) are accurate to within 0.00001?

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 7

Approximate Integration

Integration Techniques

Campbell University

Oregon State University

University of Michigan - Ann Arbor

Boston College

Lectures

01:53

In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.

27:53

In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.

26:26

(a) Find the approximation…

28:25

$\begin{array}{l}{\text { …

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05:14

05:01

11:52

27:57

12:30

03:57

15:29

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'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's ah, second drag kid. Oh, since the function is quite easy, we can't know that the second directive is also side axe. But there's a man. It's about there is a negative, but it doesn't matter. So Kay is always equal to what? And the force directive off. If it's just, I ax So the the maximum age also just what we can use this. Uh, okay. Times high. Okay, let'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's a good 10 right? And yeah, it's just Hi. No. Oh, this is 24. Sorry. 24 times. And yes, it's just hi to defy and divided by what'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'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'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'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's quicker than I o a well nigh. Am I Correct? Yeah, I'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?

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