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To verify Green's Theorem, you'll want to:

1. State Green's Theorem: $\oint_C (P\,dx + Q\,dy) = \iint_R \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA$ where $C$ is a positively oriented, simple closed curve, and $R$ is the region bounded by $C$.

2. Given Functions: Identify $P(x, y)$ and $Q(x, y)$ from the line integral.

   • $P(x, y) = y^2$
   • $Q(x, y) = x^2$

3. Compute the Partial Derivatives:

   • $\frac{\partial Q}{\partial x} = 2x$
   • $\frac{\partial P}{\partial y} = 2y$

4. Apply Green's Theorem: $\iint_R (2x - 2y) \, dA$ Evaluate this double integral over the region $R$.

5. Calculate the Line Integral:

   • Parametrize the curve $C$ and directly evaluate the line integral $\oint_C y^2 \, dx + x^2 \, dy$.

Finally, check if both sides of Green's Theorem match.

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