Chapter 6.8, Problem 391E

For the following exercises, use a computer algebraic system (CAS) and the divergence theorem to evaluate surface integral ${\displaystyle {\int }_{s}F\cdot ndS}$ for the given choice of F and the boundary surface S. For each closed surface, assume N is the outward unit normal vector.

391. Use the divergence theorem to compute flux integral ${\displaystyle {\iint }_{s}F\cdot dS}$ , where $F\left(x,y,z\right)=yj-zk$ and S consists of the union of paraboloid $y=x^2+z^2$ , $0\le y\le 1$ , and disk $x^2+z^2\le 1$ , $y=1$ , oriented outward. What is the flux through just the paraboloid?