Find the surface integral
$$\iint_S F(x,y,z)$$ if $F(x,y,z)=x+y+z$ and $S: x^2 + y^2 = 1 , 0 \leq z \leq 2$.
Obviously, this is the first type of surface integral since the function $F$ is scalar field, also, surface given is a piece of cylinder whose height is two, and this actually isn't problem at all, the problem occurs once i try to implement this.
By definition first type of surface integral is given as $$\iint_S F(x,y,z)=\iint_DF(x,y)\sqrt{1+(\frac{\partial f}{\partial x})^2+(\frac{\partial f}{\partial y})^2}dxdy$$
We don't actually have to eliminate $z$ variable every time, it actually differs from one case to another, anyway, i am having trouble with expressing $z$ in terms of two other variables since $z$ is not dependent on $x$ and $y$, i cannot express $x$ in terms of $y$ since $y=\sqrt{1-x^2}$ and $y=-\sqrt{1-x^2}$ so it gives me two distinct functions, same works for $y$ variable.
When i saw i could not do anything i mentioned above i tried introducing cylindric coordinates but then i got $$S: r=1, 0\leq z \leq 2$$ i could not go further. Anyone knows how to solve this?
Hint: Let a surface $S$ has vector equation $$\vec{r}(u,v)=x(u,v)\vec{i}+y(u,v)\vec{j}+z(u,v)\vec{k}$$$(u,v)\in D$ where $D$ is the parameter domain.Then $$\iint_S f(x,y,z)dS=\iint_Df(\vec{r}(u,v))|\vec{r_u} \times\vec{r_v}|dA$$