Let $R \subset \mathbb R^n$ a sett? (I don't know the English word for a Cartesian product of real intervals $[a,b] \times [c,d] \times \cdots$, it represents a rectangle if $n=2$ or see this question https://math.stackexchange.com/questions/1911352/english-word-for-a-pavé-in-french) and $\hat{R} \subset \mathbb R^n$ another sett such that $R \subset \text{Int}(\hat{R})$. Let $f : R \to \mathbb R$ a bounded function and lets consider the zero extension of $f$ on $\hat{R}$ defined by : $\hat{f} : \hat{R} \to \mathbb R$ : $\hat{f}(x) = f(x)$ if $x \in R$ and $\hat{f}(x) = 0$ if $x \in \hat{R} \backslash R$.
We want to prove that $$f \text{ is Riemann-integrable on } R \iff \hat{f} \text{ is Riemann-integrable on } \hat{R}$$
I was giving the general shape of the proof : taking $R_s$ a small sett around $R$ such that $Vol(R_s)-Vol(R) < \varepsilon$ and to calculate Riemann sums but I would like to write it in order to have a rigorous proof to the proposition.