What's the difference between the Riemann & Darboux Integrals?

In summary, Riemann integrals and Darboux integrals have different definitions, but are equivalent. Darboux integrals have two types: upper and lower, which depend on whether the interval function can be integrated. If both exist and are equal, the function can be integrated in the Riemann sense. Therefore, Darboux integrals encompass more functions than Riemann integrals.
  • #1
swampwiz
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I was reading about this, and they seem the same. Of course, if they were the same, they wouldn't have different names.
 
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  • #3
Riemann integrals and Darboux integrals have different definitions. However they are equivalent.
 
  • #4
One of the two authors, Frederic Riesz, describes the question in detail in points 10 and 13 of Chapter I of the book by F. Riesz and B. Sz. Nagy, Leçons d'analyse functionelle.

There are two types of Darboux-integrals: upper and lower. These integrals belong to two interval functions

$$(\beta-\alpha) \sup_{\alpha \leq x \leq \beta} f(x), ~~~~~ (\beta-\alpha) \inf_{\alpha \leq x \leq \beta} f(x). $$
If the first interval function can be integrated, then we are talking about the upper Darboux-integral, if the second, then the lower Darboux-integral. Maybe both exist, perhaps just one, possibly none.

In cases where both exist and are even equal, it is said that the function can be integrated in the Riemann sense. So Darboux's integrals have much more functions than Riemann's integrals.
 

1. What is the Riemann Integral?

The Riemann Integral is a method of calculating the area under a curve by dividing the area into smaller rectangles and summing their areas. It is named after the mathematician Bernhard Riemann.

2. What is the Darboux Integral?

The Darboux Integral is another method of calculating the area under a curve, also known as the upper and lower sums. It is named after the mathematician Jean Gaston Darboux.

3. What is the main difference between the Riemann and Darboux Integrals?

The main difference between the Riemann and Darboux Integrals is the method of partitioning the area under the curve. Riemann uses equal width rectangles while Darboux uses unequal width rectangles.

4. Which method is more accurate?

Both methods are equally accurate in calculating the area under a curve. However, Riemann is more commonly used due to its simplicity and ease of calculation.

5. Can the Riemann and Darboux Integrals be used interchangeably?

Yes, in most cases the Riemann and Darboux Integrals can be used interchangeably as they both result in the same value for the area under a curve. However, there are some functions where the two methods may give different results.

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