Answer
$2 \ln 3$
Work Step by Step
Here, we have: $Jacobian =\begin{vmatrix} \dfrac{\partial x}{\partial u}&\dfrac{\partial x}{\partial v}\\\dfrac{\partial y}{\partial u}&\dfrac{\partial y}{\partial v}\end{vmatrix}=\begin{vmatrix} 1/v&-u/v^2\\0& 1\end{vmatrix}=\dfrac{1}{v}$
$\iint_R xy dA=\int_1^{3} \int_{u^{1/2}}^{(3u)^{1/2}}1 [u \cdot (1/v)] dv du$
or, $=\int_1^3u[\ln v]_{u^{1/2}}^{(3u)^{1/2}} du$
or, $=\int_1^3u[\ln 3^{1/2}]du$
or, $=[\ln 3^{1/2}][\dfrac{u^2}{2}]_1^3$
Thus, we have
$\iint_R xy dA=\dfrac{1}{2} \ln 3 [\dfrac{9}{2}-\dfrac{1}{2}]=2 \ln 3$
