Answer
$\frac {e^{6}-7}{2}$
Work Step by Step
Since, $R=(r,\theta) | 2\leq r\leq 4, 0\leq \theta \leq \pi$
$\int\int_{R} ye^{xy}dA=\int_{0}^{3}\int_{0} ^{2}ye^{xy}dxdy$
$=\int_{0}^{3}[e^{xy}]_{0} ^{2}dy$
$=\int_{0}^{3}[e^{2y}-1]dy$
$=\frac {1}{2}[e^{2y}-y]_{0} ^{3}$
$=[\frac {e^{6}}{2}-3]-\frac{1}{2}$
$=\frac {e^{6}-7}{2}$
