Evaluate $\iint y e^{x y} d A$ for the region $R=\{(x, y) \mid 0 \leq x \leq 1,1 \leq y \leq 2\}$. A. $e^{2}+2 e-1$ B. $e^{2}+e-1$ C. $e^{2}-e-1$ D. $e^{2}+e+1$

Step 1 ：We are given the double integral \(\iint y e^{x y} d A\) over the region \(R=\{(x, y) \mid 0 \leq x \leq 1,1 \leq y \leq 2\}\).

Step 2 ：To solve this, we first integrate with respect to x while keeping y constant. This gives us \(\int_{0}^{1} y e^{x y} dx = e^{y} - 1\).

Step 3 ：We then integrate this result with respect to y. This gives us \(\int_{1}^{2} (e^{y} - 1) dy = -e - 1 + e^{2}\).

Step 4 ：Thus, the final answer is \(e^{2}+e-1\).