Answer
$4 \sqrt {22}$
Work Step by Step
We have $A(S)= \iint_{D} |\dfrac{\partial r}{\partial u} \times \dfrac{\partial r}{\partial v}|$
and $\iint_{D} dA$ is the area of the region $D$
Now, $\dfrac{\partial r}{\partial u} \times \dfrac{\partial r}{\partial v}=\lt 3,2,3\gt $
and $|\dfrac{\partial r}{\partial u} \times \dfrac{\partial r}{\partial v}|=\sqrt {3^2+2^2+3^2}=\sqrt {22}$
Therefore, $A(S)= \iint_{D} |\dfrac{\partial r}{\partial u} \times \dfrac{\partial r}{\partial v}|=\int_{-1}^1 \int_0^2 \sqrt {22} du dv$
or, $=\int_{-1}^1 [\sqrt {22} u]_0^2 dv$
or, $=[2 \sqrt {22}]_{-1}^1$
or, $=4 \sqrt {22}$
