Answer
a) Positive
b) Negative
Work Step by Step
(a) Along the $x=-3$, the vectors of $\overrightarrow{F}$ have positive y-components so since the path goes upward, $\overrightarrow{F}\cdot \overrightarrow{T}$ is always positive because the angle is acute.
This implies that $\int_{C_{1}}F\cdot dr$ = $\int_{C_{1}}F\cdot T dt=\int_{C_{1}}|F||T| \cos \theta dt$ is positive.Thus, the line integral is positive.
(b) All non-zero field vectors along the circle with radius 3 are pointed in the clockwise direction, opposite the direction to the path and hence, makes an obtuse angle with the vector field . Therefore $\overrightarrow{F}\cdot \overrightarrow{T}$ is negative and $\int_{C_{2}}F\cdot dr$ = $\int_{C_{2}}F\cdot T dt=\int_{C_{1}}|F||T| \cos \theta dt$ is negative.Thus, the line integral is negative.
