Problem $1.$ Consider the vector field $F=($:$\frac{1}{2}{x}^2,y$:$),$ the top half of the circle with radius $2,$ and the line segment from $(-2,0)$ to $(2,0)$ as pictured below. $($Note that these curves are parameterized in the counter$-$clockwise direction$)$

a$)$ Set up $($but do NOT evaluate$)$ a line integral$($s$)$ that computes the amount of work done by F on the curve$/$particle system represented by the $r(t)$ you get from the line segment. Your answer should be entirely in terms of $t.$

b$)$ Evaluate the line integral you derived in part $($a$).$

c$)$ Consider the curve joining the segments at $(-2,0)$ and $(2,0).$ State $($but do not prove$)$ the condition on $F-$ why these two different paths could give us two different values for the line integrals.