Final answer:
To evaluate the line integral, you can use the fundamental theorem of line integrals which states that if F is a vector field and C is a smooth curve, then the line integral of F along C can be calculated using a specific formula. In this case, you can parameterize the line segment from A to B and substitute the parameterization into the line integral formula.
Step-by-step explanation:
To evaluate the line integral ∮C F · dr, where C is the line segment from A to B, we can use the fundamental theorem of line integrals. This theorem states that if F is a vector field whose components have continuous partial derivatives on an open region in space and C is a smooth curve parameterized by r(t) = (x(t), y(t), z(t)) for a ≤ t ≤ b, then the line integral of F along C is given by:
∮C F · dr = ∫a to b F(r(t)) · r'(t) dt
In this case, the line segment from A to B can be parameterized as r(t) = (x(t), y(t)) where a = 0 and b = 2. Substitute the parameterization into the line integral formula and evaluate the integral using provided vector field F.