Chapter 6.7, Problem 354E

For the following exercises, use Stokes’ theorem to evaluate ${\displaystyle {\iint }_s\left(curlF\cdot N\right)dS}$ for the vector fields and surface.

354. Use the surface integral in Stokes’ theorem to calculate the circulation of field F, $F\left(x,y,z\right)=x^2{y}^{3}i+j+zk$ around C, which is the intersection of cylinder $x^2+y^2=4$ and hemisphere $x^2+y^2+z^2=16$ , $z\ge 0$ , oriented counterclockwise when viewed from above.