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(a) Find the approximations $ T_8 $ and $ M_8 $ for the integral $ \displaystyle \int_0^1 \cos (x^2)\ 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?

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a.$T_{8}=.902333$$M_{8}=.905620$b. $\left|E_{T}\right| \leq \frac{1}{128} \approx 0.0078125 \quad$ and $\quad\left|E_{M}\right| \leq \frac{1}{256} \approx 0.00390625$c. $\left|E_{T}\right| : \quad n=71$$\left|E_{M}\right| : \quad n=50$

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 7

Approximate Integration

Integration Techniques

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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.

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(a) Find the approximation…

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$\begin{array}{l}{\text { …

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problem. 19 party is quite easy because it's just calculation. We have the function to have the B A N is equal to eight. Right? And he's able to eight so we can calculate t eight. And I am eight through this formula here and here. Right? Just calculate and you can't get out, sir, On part B is to mate. Estimate the arrows. Okay, We have a formula here, Terrebonne. So we need to find the upper bone off the second director of back. Right. Let's start access. You go to call sign, I swear. But so I prime is able to two acts times sign negative, Cy Axe. Where and second directive is to minus two times Cy. I'm square clothes. Thanks. Pains to axe times cause I X square. So the abs absolute off burial is this is smaller, then. Science axes. What are the one two acts since axes for the one and quicker than zero. So this is 22 times, two times one times. This is one of the one. It's ah, six. Right here we get. The case is equal to six and we have this formula and you have the formula. Casey go six b A and they're all knowing. So the e t you is what are then six times one right? Guided by 12 times Age square, there's just one or 100. Okay, Yeah. You don't have to calculate am again because you know that I am is just ah, half off you cheap. So it's smaller, then 256 right? Okay, here's Parsi. How long do we have to choose in? Oh, okay. Used formula. Hey, six times one already by its wow times and square is Mari Ko 10. Since I wasn't right, um, to square it is greater than five. Sudden. Oh, there's no to here And there's quick witted and 71 because he's a ***. Right on the second while is just two in the square is acquitted and 5000 seems there's a two over here And that's where is greater than to Southern 500 they're squared And 50. Here's Alyssa, right? Yeah,

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