Riemann integral is zero for certain sets

[ianchenmu]

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

 

[Fredrik]

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.