= Solution
A convenient choice for this irregular region is <Monte Carlo integration>. The region is contained in $[-1,1]^2$. For independent uniform points $U_j$ in that square, let
$$
\widehat I_N=\frac4N\sum_{j=1}^N\mathbf1_{\{U_j\in R\}}\exp\!\left(e^{(U_j)_1}+e^{(U_j)_2}+e^{(U_j)_1(U_j)_2}\right).
$$
The <indicator random variable> incorporates the curved boundary directly. This estimator is unbiased; the integrand is bounded on the square, so its <variance> is finite. The <variance of a sum> of independent terms gives <root mean square> error \b[$O(N^{-1/2})$] in <big O notation>. This is a stochastic error rate. Applying a rectangular <Simpson's rule> to the indicator-weighted integrand would lose the smoothness required for its fourth-order error bound.
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