# Riemann integral is zero for certain sets

1. Mar 31, 2013

### ianchenmu

1. The problem statement, all variables and given/known data
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 $\int_{E}f(x)dV=0$.

2. Relevant equations

n/a

3. 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 $\;\;\;\;(L)\int_{E}fdV=(U)\int_{E}fdV$)

2. Mar 31, 2013

### Fredrik

Staff Emeritus
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.

Last edited: Mar 31, 2013