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Use (a) the Midpoint Rule and (b) Simpson's Rule to approximate the given integral with the specified value of $ n $. (Round your answers to six decimal places.) Compare your results to the actual value to determine the error in each approximation.

$ \displaystyle \int_0^2 \frac{x}{1 + x^2}\ dx $ , $ n = 10 $

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a.$M_{10} \approx 0.806598,$ error $\approx-0.001879$b.$M_{10} \approx 0.806598, \quad$ error $\approx-0.000060$

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 7

Approximate Integration

Integration Techniques

Campbell University

Harvey Mudd College

University of Nottingham

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.

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Okay. Problem. Fire. To solve this problem you only need is a calculator. First you need to catch the eye is equal to 0.80 or Simon one. It's adjusting to grow. And you want to die. You on. You want to calculate Am having people to Ah, 3.1 US 7.3. Right. And I started that two at 1.9. Right? He's the last off interval ways to derided by 10. Yeah, this is a And what about the ass? Us? I copy that seems a rule. So their tax is 0.2, right? Because it is to minus zero, divided by 10 Axiron is zero X one is, uh, 2.0 point two. Right ni is one point eight Just more by all the saints to better. And you will get the ass after a calculation. Um, is equal to year old 0.80 sakes by NYU zero 65 Aye. And as is he go to for 779 And what about the estimate? And what about the era? Era is I minus, um that I am And it was yes, just I meant it's ours and you get the correct answer

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