Final answer:
To evaluate the given line integral using Green's theorem, one must calculate the double integral over the region D, using the partial derivatives of the functions involved in the integral. The partial derivatives are 1 for N with respect to x and 6x for M with respect to y. The integral cannot be computed without additional information about the bounds of region D.
Step-by-step explanation:
The student is asking to evaluate the line integral using Green's theorem. Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C. Specifically, it states that the line integral ∫C M dx + N dy is equal to the double integral over D of the partial derivative ∂N/∂x minus ∂M/∂y dA, provided that the vector field is defined and has continuous partial derivatives on an open region that contains D. To apply Green's theorem to the given line integral ∫C 6xy dx + (x - y) dy, we need to identify the functions M(x, y) = 6xy and N(x, y) = x - y. Then, we calculate the partial derivative of N with respect to x, which is 1, and the partial derivative of M with respect to y, which is 6x. Green's theorem tells us to evaluate the double integral over D of ∂N/∂x - ∂M/∂y dA. This gives us the double integral over D of 1 - 6x dA. If the region D is explicitly defined and we know its bounds, we can continue to the actual computation of the integral. Unfortunately, the question does not provide a specific description of region D, so we cannot compute the integral further without additional information. If region D were, for example, a rectangle with bounds a ≤ x ≤ b and c ≤ y ≤ d, the integral would become ∫∫_D (1 - 6x) dy dx, which could be evaluated directly.