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Redo Exercise 13 with $n=100$ rectangles.

(a) 20.288350(b) 20.37835(c) 20.333325

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 4

Approximation of Areas

Integrals

Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

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.

04:04

Area Repeat Exercise $61,$…

02:30

Area Consider the sequence…

01:50

If the pattern of rectangl…

01:03

In Exercises 9 and 10 , us…

01:18

Let $A$ represent the area…

for this problem we are considering though some interval. Let's call it. Okay. All right, this will be the same penetrable from one over N plus one, two one over N. And notice that when X equals one over N plus one. The value of why there's a different color. Probably nobody here would be mm hmm. One over and respond. And The value of Y at one over N is going to equal to why it goes to one overhand. Notice that So there will be a trap destroyed there. Perhaps we can make it clear or two enlarge this one a little bit more. Perhaps one of her and is a little larger, something like this. 5-1 of the end. And that these grades attracts see a trap or something. And the area for this traffic. So it's about the qualities of area of the traffic side with equal truth and this is pretty much something that we need. Um This is just one of the traffic goods is going to be has times Bass, the first bass, we want sp two times the height. So in our case it's going to be 1/2 the first base here. We can treat them as One over n plus one. And the second date we can treat this past one of the end. The height here is a difference of the X files here, which in this case it's just going to be one over in -1 over in it is the area of the anti trump sort. Now using this result pretty much Um we can keep on writing this one awesome one half times one over n squared Finals, one over n. postman square it using this result, the total area. So total yeah is going to equal to the sound of the areas of these Protestants And equals 1 to infinity. That's what 20 lead off this event that he is. The sum from N equals one to infinity of one half. I was one of her integrated -1 over enclosed square. Now we can always pull out the constant one outside of the song. So one half times the sum from N equals one to infinity of one of the inspiration -1 over in square. This song this series here is a telescoping students. Yes, tell us jobs. It was called being serious. So we can write out the terms couple of terms and have an idea of what this one would be. For example, one half times when you plug in one, we're going to have one number one square -1. We choose one and it goes to we're going to have one of the truth break -1/3. And when we plug in the next time we're going to have one over three squared -1/4. And so when I saw what I saw, but as you can see one over n squared um 1/2 squared will cancel out one of the three squared, cancel out foursquare every single one of the terms besides the first one will cancel out this song. Here will be about half times, which is simply what happened. So that would be the two of them.

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