Answer
$4$
Work Step by Step
First, we can rewrite the integral as follows
$$\int\int_R(2+x^2y)dA=\int\int_R2dA+\int\int_Rx^2ydA$$
Now, the integral $\int\int_R(2+x^2y)dA=2(1-0)(1-(-1))=4.$ The integral $\int\int_Rx^2ydA=0$. We know this because $x^2(-y)=-x^2y$, and because of symmetry, the (negative) signed volume of the region below the $xy$-plane where $-1 \leq y \leq 0\pi$ cancels with the (positive) signed volume of the region above the
xy-plane where $0 \leq y \leq1$.
Thus, result is $4$.
