# Calculate the area of the surface parameterised form

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