Answer
$\frac{1}{2}(e^2-3)$
Work Step by Step
$\int\int_R xy e^{xy^2}dA$
=$\int_0^2\int_0^1 xy e^{xy^2}dydx$
=$\int^{ln2}_{0}[\frac{1}{2}e^{xy^2}]^{ln2}_0$dx
=$\int^{ln2}_{0}[-e^{x}]^{1}_0$dx
=$\int^{2}_{0}(\frac{1}{2}e^x-\frac{1}{2})dx$
=$[\frac{1}{2}e^x-\frac{1}{2}x]^2_0$
=$\frac{1}{2}(e^2-3)$
