Chapter 6.8, Problem 390E

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.

390. Use the divergence theorem to compute the value ${\displaystyle {\iint }_{s}F\cdot dS}$ , where $F\left(x,y,z\right)=\left(y^3+3x\right)i+\left(xz+y\right)j+\left[z+x^4\cos \left(x^{2}y\right)\right]k$ and S is the area of the region bounded by $x^2+y^2=1$ , $x\ge 0$ , $y\ge 0$ , and $0\le z\le 1$ .