Problem

乙、計算、證明題：共 3 題，每題 12 分，共 36 分。須詳細寫出計算及證明過程，否則不予計分。

2. Evaluate $\displaystyle\iint_R xe^{y^2}\,dA$ where $R$ is the plane region bounded by the $y$-axis, $x=0$, the horizontal line $y=4$, and the graph of $y=x^2$.