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위키데이터

• ID : Q697181

말뭉치

1. In real analysis, Riemann Integral, developed by the mathematician Bernhard Riemann, was the first accurate definition of the integral of a function on an interval.
2. Riemann integral is applied to many practical applications and functions.
3. Although having such huge applications, the Riemann integral is quite challenging to handle as we can see that its definition is little sophisticated.
4. So, it is a bit inconvenient to use Riemann integral in practical life.
5. And how this is used to define the Riemann integral.
6. The Riemann integral is defined in terms of Riemann sums.
7. In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval.
8. In educational settings, the Darboux integral offers a simpler definition that is easier to work with; it can be used to introduce the Riemann integral.
9. The Darboux integral is defined whenever the Riemann integral is, and always gives the same result.
10. Loosely speaking, the Riemann integral is the limit of the Riemann sums of a function as the partitions get finer.
11. Since the Riemann integral is related to the area under the graph of f, the only important information is the shape of the graph.
12. The Riemann integral is the definite integral normally encountered in calculus texts and used by physicists and engineers.
13. As an example of the application of the Riemann integral definition, find the area under the curve from 0 to .
14. Other geometric and physical quantities, such as volume and work, fit easily into the framework supplied by the concept of the Riemann integral.
15. Moreover a function which possesses a Riemann integral must exhibit a great deal of regularity.
16. The need for regularity means that the convergence theorems for the Riemann integral are severely restricted.
17. As in chapter FIXME, we define the Riemann integral using the Darboux upper and lower integrals.
18. We now have all we need to define the Riemann integral in \(n\)-dimensions over rectangles.
19. Again, the Riemann integral is only defined on a certain class of functions, called the Riemann integrable functions.
20. If so, find the value of the Riemann integral.
21. Now we can state some easy conditions that the Riemann integral satisfies.
22. All of them are easy to memorize if one thinks of the Riemann integral as a somewhat glorified summation.
23. Note that this theorem does not say anything about the actual value of the Riemann integral.
24. We start by describing the Riemann integral, which is commonly taught in elementary calculus, and then describe the relationship between integration and differentiation.
25. The next section covers the Lebesgue integral, which is technically harder than the Riemann integral and requires measure theory.
26. follow from the definition of the Riemann integral.
27. where p > 1 ) instead of a mesh (or the Riemann norm) to derive some equivalences of the Riemann integral.
28. To begin with, the usual settings of a Riemann integral are listed for comparison.
29. Since the p -norm is massively used in functional analysis, by defining an alternative Riemann integral via this norm, one could further look at the typical the Riemann integral from a new aspect.
30. This might further extend the Riemann integral to other territories.

메타데이터

위키데이터

• ID : Q697181

Spacy 패턴 목록

• [{'LOWER': 'riemann'}, {'LEMMA': 'integral'}]