Chapter 13.5, Problem 47PS

To determine

To calculate: The value of the integral ${\displaystyle \underset{S}{\iint }\overrightarrow{\text{F}}\cdot \overrightarrow{\text{N}}}dS$ for the vector field $\overrightarrow{\text{F}}=x^2\widehat{i}+z\widehat{k}$ and $S$ is the parametric surface $x=\sin u\cos v$ , $y=\sin u\sin v$ , $z=\cos u$ for $0\le u\le \pi ,\text{ 0}\le v\le 2\pi$ .