1. The problem asks to determine the sign (positive, negative, or zero) of the line integral of given vector fields along specified oriented paths. 2. Recall that the line integral of a vector field \( \mathbf{F} \) along a curve \( C \) is given by: $$\int_C \mathbf{F} \cdot d\mathbf{r}$$ This integral measures the work done by the vector field along the path. 3. Important rules: - If the vector field is tangent and in the same direction as the path, the integral tends to be positive. - If the vector field is tangent but opposite to the path direction, the integral tends to be negative. - If the vector field is perpendicular to the path or the path is closed and the vector field is conservative, the integral can be zero. 4. Analyze Graph 1 (top-left): - Vector field rotates counterclockwise around the origin. - Path is a spiral starting from positive x-axis, going counterclockwise inward. - The vector field direction aligns with the path direction (both counterclockwise). - Since the path moves inward but follows the vector field rotation, the line integral is positive. 5. For the other graphs, since the question asks only for the first problem, we do not analyze them here. Final answer: The line integral for Graph 1 is positive.