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Use the Midpoint Rule with the given value of $ n $ to approximate the integral. Round the answer to four decimal places.

$ \displaystyle \int^2_0 \frac{x}{x + 1}\, dx $, $ n = 5 $

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00:30

Frank Lin

00:55

Amrita Bhasin

Calculus 1 / AB

Chapter 5

Integrals

Section 2

The Definite Integral

Integration

Missouri State University

Campbell University

Harvey Mudd College

Idaho State University

Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

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Use the Midpoint Rule with…

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03:20

Yeah. So in the problem you see here, problem number 11, we have a definite integral and were asked to estimate this using uh rebound sums and rectangles, with the number of rectangles being five. So a crude approximation. So over the interval from 0 to 2, So two divided by five, his point four. So when I divide this up, you're going to have 0.4. Yeah 0.8 1.2, 1.6. Now were also asked to use the midpoint rule. So for mid points, the mid points are going to be a point to 0.6. Um 1.0, 1.4 and 1.8. So this integral From 0 to 2 of X over X plus one dx can be approximated by The areas of these rectangles. So each rectangle is a width .4, so 0.4 times. And now we have to figure out so it's going to be substituting each of these mid points into this formula. So it's going to be 0.2 over um actually just gonna be 1.2. So let me just do it that way, one plus 10.2, so point to over 1.2 Plus .6 over 1.6 plus one over to Plus 1.4 Over 2.4 plus 1.8 Over 2.8. So that is going to be my area approximation, so I can just do all of that on my calculator. So it is .4 times. And so we had what .4 divided about 1.4 Plus .8, divided by 1.8 plus um one divided by two, so one half Plus 1.4 Over 2.4 plus 1.8, divided by 2.8 98254 is approximation, so 98254 is my approximation. Yeah. And we can see how accurate that is. We'll find better ways of doing it ourselves. But I mean what if we were to ask for? Yeah. The integral From 0 to 2 of X over X plus one, So .90139. So I'm in the ballpark, but not a great approximation.

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