If instead of evaluating the above line integral in counter-clockwise direction, I evaluate it via the clockwise direction, would that change the answer? What if I evaluate $C_1$ and $C_3$ in the counter-clockwise direction, but I evaluate $C_2$ in the clockwise direction?
If the direction does matter, in which direction would I evaluate the below line integral?
Best Answer
Direction does not matter for the line integral of a function, but here you are dealing with a work integral (i.e. the integral of a vector field along the curve). In the latter case, orientation does matter.
The statement of Green's Theorem includes (or it should, to make sense) the orientation required for the equality to hold. The orientation for the curve is the one that leaves to region $R$ to your left as you traverse the curve.