Green's theorem on a trigonometric curve

Question

I'm reviewing for an exam, and picked this problem out for practice:

Use Green's theorem to evaluate $$\oint_C y^2dx+xdy \,$$

For the curve $C$, $$C = \alpha(t) = 2\cos^3(t) \hat{i} + 2\sin^3(t) \hat{j}$$

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Here's my work so far, the bounds have me stumped.

Answer 1

Here's the solution I came up with. If I understand correctly the theorem also provides this determinant that gives the set of points found in the 3rd line of the first pic.