Final answer:
To evaluate the line integral ∫Cx²ydx−xy²dy using Green's Theorem, we need to find the curl of the vector field F(x, y) = (Cx²y, -xy²), then find the area enclosed by the given ellipse x² + 2y² = 2, and finally integrate over that region to obtain the answer.
Step-by-step explanation:
To evaluate the line integral ∫Cx²ydx−xy²dy using Green's Theorem, we first need to find the curl of the vector field F(x, y) = (Cx²y, -xy²). The curl is given by ∇ × F = (∂(-xy²)/∂x) - (∂(Cx²y)/∂y). Evaluating the partial derivatives and simplifying, we get ∇ × F = -2Cx² - 2xy.
Next, we need to find the area enclosed by the ellipse x² + 2y² = 2. We can rewrite this equation as y = sqrt((2 - x²)/2) to express y in terms of x. Now, we can use Green's Theorem to evaluate the line integral over the ellipse by applying the formula ∫∫(∇ × F) dA = ∫∫-2Cx² - 2xy dA, where dA is the infinitesimal area element.
Finally, we integrate over the region enclosed by the ellipse by substituting the limits of integration, which are determined by the curve of the ellipse. This will give us the final result of the line integral.