$(10$ points$)$ Let vec$(C)be$ the curve defined $byy=1-x^2,$ oriented counterclockwise. Use Green's

Theorem $to$ evaluate the line integral

where the vector field $is$ given $by$

${\displaystyle \underset{\text{v}\text{e}\text{c}(C)}{\overset{\phantom{}}{}}}$vec$(F)*dvec(r)$

vec$(F)(x,y)=($:$-y+e^x,2x-e^{y^2}$:$)$

${\displaystyle \underset{c}{\overset{\phantom{}}{}}}(-1,0)$

Hint: The curve vec$(C)is$ not closed. $To$ apply Green's Theorem, you must first close the curve

appropriately.