Let $B =$ {$(x,y) \in \mathbb{R}^2 : x^2 + y^2 \leq 1$}. Consider $S=$ {$(x,y,z) \in \mathbb{R}^3 : z = x^2 + y^2 ; x^2 + y^2 \leq 1$} over $B$, oriented so that the function $\alpha: B \to S$ defined by $\alpha = (x,y,x^2 +y^2)$ is orientation-preserving. Identify the boundary of S ($\partial S$) and compute:
$\int_{\partial S}$ $(cos(z))dx + (xz + tan(y))dy + y^2z^3dz$.
I'm having trouble trying to identify the boundary of $S$. $S$ appears to be the intersection of the solid unit rod with the paraboloid $z = x^2 + y^2$.
So would the boundary of $S$ be the surface of paraboloid for $0 \leq z \leq 1$
If so, would it be appropriate to apply Stokes Theorem? But how do I identify the unit normal?
$S$ is just the surface of the paraboloid, not a three dimensional shape as you seem to be thinking. The boundary is the circle $\{(x,y,1):x^2+y^2=1\}$.
You can evaluate your integral directly now by parameterizing this circle. Or you could use Stoke's theorem to turn this into a surface integral, but that is probably harder.