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Let's solve this step-by-step using **Green's theorem**. 

### Problem Restated:
We are asked to evaluate the line integral  
\[
I = \oint_C \left( y^3 \, dx - 2x^3 \, dy \right),
\]  
where \( C \) is the **counterclockwise-oriented circle** \( x^2 + y^2 = a^2 \).

---

### Step 1: Applying Green's Theorem
Green's theorem states:
\[
\oint_C \left( M \, dx + N \, dy \right) = \iint_R \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) \, dA,
\]
where \( C \) is a positively-oriented, simple closed curve, and \( R \) is the region enclosed by \( C \). In our case:
- \( M(x, y) = y^3 \)
- \( N(x, y) = -2x^3 \)

---

### Step 2: Partial Derivatives
We now compute the required partial derivatives:
\[
\frac{\partial N}{\partial x} = \frac{\partial (-2x^3)}{\partial x} = -6x^2,
\]
\[
\frac{\partial M}{\partial y} = \frac{\partial (y^3)}{\partial y} = 3y^2.
\]

---

### Step 3: Evaluating the Double Integral
Green’s theorem converts the line integral into the following double integral over the disk \( x^2 + y^2 \leq a^2 \):
\[
I = \iint_R \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) \, dA = \iint_R \left( -6x^2 - 3y^2 \right) \, dA.
\]

In polar coordinates:
- \( x = r \cos \theta \),
- \( y = r \sin \theta \),
- \( dA = r \, dr \, d\theta \),
- The region \( R \) is described by \( 0 \leq r \leq a \) and \( 0 \leq \theta < 2\pi \).

Thus, the integrand becomes:
\[
-6x^2 - 3y^2 = -6r^2 \cos^2 \theta - 3r^2 \sin^2 \theta.
\]
Factor out \( r^2 \):
\[
-6r^2 \cos^2 \theta - 3r^2 \sin^2 \theta = -r^2 \left( 6\cos^2 \theta + 3\sin^2 \theta \right).
\]

Using the given trigonometric identities:  
\[
\cos^2 \theta = \frac{1 + \cos(2\theta)}{2}, \quad \sin^2 \theta = \frac{1 - \cos(2\theta)}{2},
\]
we get:
\[
6 \cos^2 \theta + 3 \sin^2 \theta = 6 \cdot \frac{1 + \cos(2\theta)}{2} + 3 \cdot \frac{1 - \cos(2\theta)}{2}.
\]

Simplify:
\[
6 \cdot \frac{1 + \cos(2\theta)}{2} = 3 + 3\cos(2\theta),
\quad 3 \cdot \frac{1 - \cos(2\theta)}{2} = \frac{3 - 3\cos(2\theta)}{2}.
\]  


By using Green's theorem, evaluate the curve integral I = ∮_C (y^3 dx - 2x^3 dy), where C is the counterclockwise circle x^2 + y^2 = a^2.

Learn how to evaluate a line integral using Green's Theorem. The problem involves the counterclockwise circle x^2 + y^2 = a^2 and covers key concepts in vector calculus, including converting to polar coordinates and applying Green's Theorem. Suitable for undergraduate students studying calculus.

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