Consider the following system of polynomial inequalities with variables $q_1,\ q_2$.
\begin{align*} a_1 \leq q_1 + q_2 \leq a_2\\ b_1 \leq q_1^2 + q_2^2 \leq b_2\\ c_1 \leq q_1^3 + q_2^3 \leq c_2 \\ 0<q_1,\ q_2 <1 \end{align*} Assume that there exists a feasible solution $(p_1, p_2)$ and let $D$ be the set of solutions of the above system.
I am not sure if this helps but we know that $a_1, a_2, b_1,\ldots$ have the following form \begin{equation} a_1 = p_1 + p_2 - \epsilon,\ a_2 = p_1 + p_2 + \epsilon \\ b_1 = p_1^2 + p_2^2 - \epsilon,\ b_2 = p_1^2 + p_2^2 + \epsilon \\ c_1 = p_1^3 + p_2^3 - \epsilon,\ c_2 = p_1^3 + p_2^3 + \epsilon \end{equation} where $0< \epsilon < 1$.
I know (through experiments) that the feasible set of this system consists of at most $2$ connected components.
Questions
Is there any method to upper bound the area of the feasible set $D$ as a function of $a_1, a_2, b_1, b_2, c_1, c_2$ ?
Could it be done analytically using integration ?