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If you have a CAS that evaluates midpoint approximations and graphs the corresponding rectangles (use RiemannSum or middlesum and middlebox commands in Maple), check the answer to Exercise 11 and illustrate with a graph. Then repeat with $ n = 10 $ and $ n = 20 $.

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00:02

Frank Lin

Calculus 1 / AB

Chapter 5

Integrals

Section 2

The Definite Integral

Integration

Missouri State University

Campbell University

Harvey Mudd College

Boston College

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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If you have a CAS that eva…

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Some computer algebra syst…

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Refer to the graph in Fig.…

So if you don't have access to Maple or some other kind of cast software that will both graph these rectangles for the midpoint as well as give you the approximation using the midpoint formula Desperate actually has a calculator for this. Um, that does both the summing, which you can see over here as well as graphing rectangles like we have our here. So the first thing we need to do is go ahead and actually plot hours. So will replace affects here. So the X over X plus one next they first want us to check will be got in number five just to make sure it's right. So I'm going to go ahead and plug five into here. Uh, and this should give us an area of actually, Before we do that, I need to change my bounds also. So it would be zero 22 So this is the region were interested. Let me zoom into this. Okay, so this is supposed to be that graph with the rectangles. That's what they were talking about. Putting the rectangles on there and then down here is our approximation. And I believe that is essentially what we have got, uh, the Pentagon, where you may have rounded for number 11 and then just to get 10 and 20 more tangled to become up here and change in so in is equal to 10. So this is the graph down here is our approximation. And then we do the same thing for 20 so graph and then our approximation.

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