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Express the limit as a definite integral.

$ \displaystyle \lim_{n \to \infty} \frac{1}{n} \sum^{n}_{i = 1} \frac{1}{1 + (i/n)^2} $

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$\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{i=1}^{n} \frac{1}{1+(i / n)^{2}}=\int_{0}^{1} \frac{1}{1+x^{2}} d x$

00:56

Frank Lin

00:48

Amrita Bhasin

Calculus 1 / AB

Chapter 5

Integrals

Section 2

The Definite Integral

Integration

Campbell University

Oregon State University

Baylor University

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.

00:07

Express the limit as a def…

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0:00

some as indefinite Any girl. The first part is one over N that's going to be basically the width of each of the rectangle, so we could go ahead and convert that to DX. Um, the next thing we might notice is that we're only going from zero upto one. And the reason we know that is because it all rests on the definition of Delta X here. So let me explain that a little bit. It's b minus a over and and because that's all equal to one of rent, then we can basically assume that be is one and a zero. So that's how we end up with zero toe one there aan den. The last thing is that this expression times I will give us the of component here, and that just becomes X. So really the integration and is 1/1 plus X squared. And that would be the rewritten form of the limit on the some from above. There

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