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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.

Would you like the full solution, step-by-step?

Learn how to verify Green's Theorem for the line integral Int(C) y^2 dx + x^2 dy. Step-by-step solution includes applying Green's Theorem, computing partial derivatives, and evaluating double integrals over the region R. Suitable for advanced undergraduate students.

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