Calculating the flux through $K=${$(x,y,z)|(z-1)^2=x^2+y^2, 0 \leq z \leq 1$} with $F=-y \hat{\imath}+x \hat{\jmath}+z \hat{k}$

Question

The surface $S \subset \Bbb R^3$ is a union of

$K=${$(x,y,z)|(z-1)^2=x^2+y^2, 0 \leq z \leq 1$}

and

$C=${$(x,y,z)|x^2+y^2 \leq 1 , z=0$}

Given is the vectorfield $F=-y \hat{\imath}+x \hat{\jmath}+z \hat{k}$

Calculate the flux through $S$ with an outward pointing normal, without using the divergence theorem.

I calculated the flux through $C$, which is $0$. Because $F \bullet \hat{N}=( -y \hat{\imath}+x \hat{\jmath}+z \hat{k}) \bullet -\hat{k}= -z \hat {k}$, but $z=0$ in $C$ so, $F \bullet \hat{N}=0$.

I'm having trouble with calculating the flux through $K$. This is because I find it difficult to find a normal vector. Tips and hints about this are appreciated !