Chapter 17.7, Problem 22E

Stokes’ Theorem for evaluating surface integrals Evaluate the line integral in stokes’ Theorem to determine the value of the surface integral ${\displaystyle {\iint }_s\left(\nabla \times F\right)}$∙n dS. Assume n points in an upward direction.

22. $\text{F}=\langle 4x,-8z,4y\rangle$; S is the part of the paraboloid z = 1 − 2x² – 3y² that lies within the paraboloid z = 2x² + y².