Riemann integral is zero for certain sets

In summary, in this conversation, the question is about proving that if a subset of a given set is a closed Jordan domain and a given function is Riemann integrable on it, then the integral of the function over the subset is equal to zero. The discussion also includes a question about how to relate the condition of being Riemann integrable to the value of the integral being zero, and a suggestion to try a simple case before the general case. The definition of a "closed Jordan domain" is also mentioned.
  • #1
ianchenmu
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Homework Statement


The question is:

Let ##\pi=\left \{ x\in\mathbb{R}^n\;|\;x=(x_1,...,x_{n-1}, 0) \right \}##. Prove that if ##E\subset\pi## is a closed Jordan domain, and ##f:E\rightarrow\mathbb{R}## is Riemann integrable, then ##\int_{E}f(x)dV=0##.




Homework Equations



n/a

The Attempt at a Solution


(How to relate the condition it's Riemann integrable to the value is ##0##? The textbook I use define ##f## is integrable on ##E## iff ##\;\;\;\;(L)\int_{E}fdV=(U)\int_{E}fdV##)
 
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  • #2
What is the definition of "closed Jordan domain"?

Regardless of what the answer to that is, the strategy here should definitely be to prove that given ##\varepsilon>0##, there's an upper sum U and a lower sum L such that ##-\varepsilon<L<U<\varepsilon##. You may want to try this for an especially simple choice of E and f before you try the general case.
 
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FAQ: Riemann integral is zero for certain sets

1. What is the Riemann integral?

The Riemann integral is a mathematical concept used to calculate the area under a curve on a graph. It was developed by mathematician Bernhard Riemann in the 19th century and is widely used in calculus and analysis.

2. How is the Riemann integral calculated?

The Riemann integral is calculated by dividing the area under a curve into smaller rectangles, calculating the area of each rectangle, and then summing them all together. As the number of rectangles approaches infinity, the Riemann integral becomes more accurate.

3. Why is the Riemann integral important?

The Riemann integral is important because it allows us to calculate the area under a curve, which is useful in many real-world applications. It also plays a crucial role in the development of calculus and other areas of mathematics.

4. Can the Riemann integral be zero for certain sets?

Yes, the Riemann integral can be zero for certain sets. This occurs when the function being integrated is equal to zero over the entire interval of integration. In other words, the area under the curve is zero.

5. What are some examples of sets where the Riemann integral is zero?

Some examples of sets where the Riemann integral is zero are functions like f(x) = 0, which is equal to zero for all values of x, and piecewise functions with segments that cancel each other out, resulting in a total area of zero.

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