Final Answer:
The line integral ∫(y²x dx + 6x²y dy) over the positively oriented square C with vertices (0, 0), (3, 0), (3, 3), and (0, 3) using Green's Theorem is 54.
Step-by-step explanation:
Green's Theorem states that for a positively oriented region R with piecewise smooth boundary C, if P(x, y) and Q(x, y) have continuous partial derivatives on an open region containing R, then the line integral of P and Q along C is equal to the double integral of (∂Q/∂x - ∂P/∂y) over the region R. In this case, P(x, y) = y²x and Q(x, y) = 6x²y.
Applying Green's Theorem, we calculate the double integral of (6x² - 2xy) over the region R, where 0 ≤ x ≤ 3 and 0 ≤ y ≤ 3. The result is ∫[0 to 3] ∫[0 to 3] (6x² - 2xy) dy dx, which evaluates to 54.
Using Green's Theorem in this context allows us to transform a line integral into a double integral over the region R. It provides a powerful tool for simplifying certain types of integrals and relates line integrals to the flux across a region. In this example, the line integral is efficiently evaluated by calculating the double integral over the specified square region, resulting in the final answer of 54.