I have to calculate the area of the surface $$ S=\{(x,y,z) \in \mathbb{R^3} :x^2+y^2 = 2, 0 \le z \le 5-2x\}$$
If I parametrize $r(t)=(\sqrt2 \cos t, \sqrt2 \sin t)$ then I cannot say that $z=f(x,y)=f(r(t))$ because $z$ varies.
What am I doing wrong in the interpretation of the exercise?
How can I proceed to calculate the line integral $\int_\gamma f(r(t)) ds $ ?
The surface S is a 2 dimension object (it's a sort of cylinder with a oblique cut ) so to describe it with parameters you need two of them.
As you said you can begin describing the bottom of this "cylinder", and that is :
$$B_S=\textbf{Image}((s\sqrt2 \cos t,s\sqrt2 \sin t,0))$$ with $t \in [0,2 \pi]$ and $s \in [0,1]$
then we can describe the lateral surface of this "cylinder", and that is
$$L_s=\textbf{Image}((\sqrt2 \cos t,\sqrt2 \sin t,(5-2\sqrt2 \cos t)s ))$$
with $t \in [0,2 \pi]$ and $s \in [0,1]$
then you have to describe the top of this "cylinder" in a similar way. Have you got a solution for this?
Then you can use a theorem for multivariable calculus to calculate the area of $\textbf{each}$ surface defined above:
Let A be a 2-surface (surface of 2 dimension) in $\mathbb{R^3}$ given parametrically by $\textbf{r}(s,t)=(r_1(s,t),r_2(s,t),r_3(s,t)) $, with $s \in [a,b]$ and $t \in [c,d]$ then $$\textbf{surface}(A)=\int_a^b\int_c^d ||\frac{\partial \textbf{r}}{\partial t}\times \frac{\partial \textbf{r}}{\partial s} || dt ds$$ where $\times$ is the cross product.
You do the sum of the areas of the three surfaces and it's done.
$\textbf{Hint}$: to calculate the area of $B_S$ you don't need to do the computations of the above formula