Using Green's Theorem, Find the Outward Flux of F Across

Multiple Choice

Using Green's Theorem, find the outward flux of F across the closed curve C.
- $\mathbf { F } = \left( \mathrm { x } - \mathrm { e } ^ { \mathrm { x } } \cos \mathrm { y } \right) \mathbf { i } + \left( \mathrm { x } + \mathrm { e } ^ { \mathrm { x } } \sin \mathrm { y } \right) \mathbf { j } ; \mathrm { C }$ is the lobe of the lemniscate $\mathrm { r } ^ { 2 } = \sin 2 \theta$ that lies in the first quadrant

A) $\frac { 1 } { 4 }$
B) 0
C) 1
D) $\frac { 1 } { 2 }$