Answer
$$4 \sqrt {22}$$
Work Step by Step
We have $A(S)= \iint_{D} |r_u \times r_v|$; 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, $$A(S)= \iint_{D} |r_u \times r_v|=\int_{-1}^1 \int_0^2 \sqrt {22} du dv\\=\int_{-1}^1 [\sqrt {22} u]_0^2 dv \\=[2 \sqrt {22}]_{-1}^1 \\=4 \sqrt {22}$$
