Chapter 6.8, Problem 394E

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.

394. Consider $F\left(x,y,z\right)=x^{2}i+xyj+\left(z+1\right)k$ . Let E be the solid enclosed by paraboloid $z=4-x^2-y^2$ and plane $z=0$ with normal vectors painting outside E. Compute flux F across the boundary of E using the divergence theorem.