Calculate the area of the surface parameterised by
$\displaystyle r(u,v):= (\frac{1}{2}u^2,uv,v^2) $,
$\displaystyle 0 \le u \le 2, -1 \le v \le 1 $
$\displaystyle r_u= (u, v, 0)$, $\displaystyle r_v= (0, u, 2v)$. The "fundamental vector product", and what you need to integrate, is the magnitude of the cross product of those two vectors, times dudv.
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