Question

Let F = xyi + yzj + xzk and C is the boundary of S, the surface z = 16 − x2 for 0 ≤ x ≤ 4 and −7 ≤ y ≤ 7, oriented upward. Use Stokes' Theorem to find the integral over the region c of F.dr

Answer 1

Stokes theorem states:
\[\int\limits_{C}F. dl=\int\limits\int\limits\int\limits_{V}FdV\]
where V is the volume enclosed by the surface. So just solve triple integral:
\[\int\limits_{0}^{4}dx\int\limits_{-7}^{7}dy\int\limits_{0}^{16-x^2}Fdz\]