Chapter 6.7, Problem 371E

For the following exercises, let $S$ he the disk enclosed by curve $C:\text{r}(t)=\langle \cos \phi \cos t,\sin t\sin \phi \cos t\rangle$ , for $0\le t\le 2\pi$ , where $0\le \phi \le \frac{\pi }{2}$ is a fixed angle.

371. Circle $C$ in plane $x+y+z=8$ has radius $4$ and center $(2,3,3)$ . Evaluate ${\displaystyle {\oint }_c\text{F}\cdot d\text{r}}$ for $\text{F}=\langle 0,-z,2y\rangle$ , where $C$ has a counterclockwise orientation when viewed from above.