Chapter 6.8, Problem 410E

For the following exercises, use a CAS and the divergence theorem to compute the net outward flux for the vector fields across the boundary of the given regions D.

410. Let E be the solid bounded by the xy-plane and paraboloid $z=4-x^2-y^2$ so that S is the surface of the paraboloid piece together with the disk in the xy-plane that farms its bottom. If $F\left(x,y,z\right)=\left(xz\sin \left(yz\right)+x^3\right)i+\cos \left(yz\right)j+\left(3z{y}^2-e^{x^2+y^2}\right)k$ , find ${\displaystyle {\iint }_{s}F\cdot dS}$ using the divergence theorem.