Upper Darboux Integral: Not Integrable?

Thread starter ergonomics
Start date Jul 13, 2006
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Integral

In summary, the Upper Darboux Integral is a method of calculating the area under a curve using rectangles and is a type of Riemann Integral that uses the upper bound of the function to determine the height of the rectangles. It differs from other types of integrals in that it only uses the upper bound and is sometimes referred to as "not integrable" for certain functions. However, it can still be useful in certain situations, such as providing a more accurate approximation of the area under a curve and calculating integrals of functions with a finite number of discontinuities.

Jul 13, 2006

ergonomics

if a function f is not integrable does it imply, that there exists such an epsilon that whichever partition P i choose it will follow this condition
Uf,p-Lf,p>=Epsilon?

Last edited: Jul 13, 2006

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Jul 13, 2006

ircdan

Looks ok yea, f:[a,b]->R not integrable implies there exists an e > 0 such that for all partitions P of [a,b], U - L >= e.

Last edited: Jul 13, 2006

Jul 13, 2006

ergonomics

Thank you.

FAQ: Upper Darboux Integral: Not Integrable?

What is the Upper Darboux Integral?

The Upper Darboux Integral is a method of calculating the area under a curve using rectangles. It is a type of Riemann Integral that uses the upper bound of the function to determine the height of the rectangles.

How is the Upper Darboux Integral different from other types of integrals?

The Upper Darboux Integral is different from other types of integrals, such as the Riemann Integral and the Lebesgue Integral, because it uses the upper bound of the function to determine the height of the rectangles instead of the lower bound or a combination of both.

Why is the Upper Darboux Integral sometimes referred to as "not integrable"?

The Upper Darboux Integral is sometimes referred to as "not integrable" because there are certain functions that do not have a defined Upper Darboux Integral. This means that the Upper Darboux Integral cannot be used to calculate the area under the curve for these types of functions.

What are some examples of functions that are not integrable using the Upper Darboux Integral?

Functions that have discontinuities, such as the step function, are not integrable using the Upper Darboux Integral. Additionally, functions that have infinite oscillations or unboundedness are also not integrable using this method.

Are there any benefits to using the Upper Darboux Integral?

While there are some limitations to using the Upper Darboux Integral, it can still be a useful tool in certain situations. For example, it can provide a more accurate approximation of the area under a curve compared to other methods. It can also be used to calculate integrals of functions that are not continuous but have only a finite number of discontinuities.