This is a follow up question to my original question: rational function with area e under it. I have stated it more generally and more clearly after discussing it. If there are any improvements to be made to how the question has been stated please make the changes.
Let A be the set of all $x \in \mathbb{R}^n$ such that $P(x) > 0$ and $\left\vert x\right\vert < r$ where $P$ is a rational function with rational coefficients, $n$ is finite and $r \in \mathbb{Q}$. Does there exist a $P$ and $r$ such that the volume of $A$ (hyper volume, n-dimensional volume, etc.) equals Euler’s constant $e$?
Prove that such a $P$ and $r$ do not exist or find such a $P$ and $r$.