> 
> Let $n$ be a positive integer, $f:\mathbb R^n\to\mathbb R$ be the map $(x\_1,\ldots,x\_n)\mapsto \prod\_{i=1}^n x\_i$, and $$\overline D\_1(0) = \left\{x\in\mathbb R^n: \sum\_{i=1}^n x\_i^2\leqslant 1 \right\}$$be the closed unit disk centered at $0$. Then $$\int\_{\overline D\_1(0)}f\ \mathsf d\lambda=0 $$ where $\lambda$ is Lebesgue measure. 
> 
> 
> 

My argument: The magnitude of the integral of $f$ over each orthant of $\mathbb R^n$ is the same, and $f$ is positive on half of them and negative on the rest, and hence summing the integrals of $f$ over each orthant yields $0$.

I verified the result for $n=1,2,3$ but I'm not entirely convinced my proof is rigorous.