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Riemann integral is zero for certain sets

ianchenmu

Member
Feb 3, 2013
74
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 $\int_{E}f(x)dV=0$.


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