Riemann Integral II Examples of Riemann integral II Real Analysis

#Real Analysis #B.Sc Non-Med #M.Sc Mathematics #B.Tech
Examples of Riemann Integral
Theorem Of Real Analysis
In this video we introduce the formal analytic theory of integration.
There are two major ways to define integration.
These are Riemann integration and Lebesgue integration.
Of these two Riemann integration is simpler and this is the one we look at here.
There are multiple equivalent ways of defining the Riemann integral.
The one we will look at in this video is called the Darboux definition.
The Darboux definition is easier to work with and prove things with than the original definition that Riemann first used, but is ultimately equivalent to the original Riemann definition.
It defines a function to be Riemann.integrable if the supremum over all dissections of the lower Riemann sums is the same as the infimum over all dissections of the upper Riemann sums. This value is then the value of the integral.
We explain and motivate this definition in the video and also give an example of proving a function is Riemann integrable over an interval and o different example of proving a function isn’t Riemann integrable over an interval.
Contents:
Part 1 - Definition of a dissection/partition
Parts 2 to 4 - Riemann Sums
Part 5 - Definition of Riemann Integration
Part 6 - Example of proving a function is Riemann Integrable

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[@silentintegrals9104]

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[@harshmarkam6883]

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[@karansirmathematics]

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[@ParshuramYadav-ww6hc]

Summation of (r-1)^3=[ n(n-1)/2]^2 how