To compute the circulation of the vector field F around the rectangle $C$, we apply Green's Theorem:

$\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_R \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dA$

where $\mathbf{F} = P \mathbf{i} + Q \mathbf{j}$, with $P = 7\cos(-3y)$ and $Q = 4x^2 \sin(-3y)$. The region $R$ is the rectangle with vertices $(0,0)$, $(2,0)$, $(0,2)$, and $(2,2)$.

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Step 1: Compute $\frac{\partial Q}{\partial x}$ and $\frac{\partial P}{\partial y}$

$\frac{\partial Q}{\partial x} = \frac{\partial}{\partial x} \left( 4x^2 \sin(-3y) \right) = 8x \sin(-3y)$

$\frac{\partial P}{\partial y} = \frac{\partial}{\partial y} \left( 7\cos(-3y) \right) = 7 \cdot (-3) \sin(-3y) = -21\sin(-3y)$

Thus, $\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 8x \sin(-3y) - (-21\sin(-3y)) = 8x \sin(-3y) + 21\sin(-3y)$ $\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = \sin(-3y)(8x + 21)$

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Step 2: Set up the double integral

The bounds of integration for $R$ are: [ x \in [0, 2], \quad y \in [0, 2] ]

The double integral becomes: $\iint_R \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dx\,dy = \int_0^2 \int_0^2 \sin(-3y)(8x + 21) \, dx\,dy$

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Step 3: Evaluate the inner integral with respect to $x$

$\int_0^2 (8x + 21) \, dx = \int_0^2 8x \, dx + \int_0^2 21 \, dx$

$\int_0^2 8x \, dx = 8 \left[ \frac{x^2}{2} \right]_0^2 = 8 \cdot 2 = 16, \quad \int_0^2 21 \, dx = 21 \cdot 2 = 42$

$\int_0^2 (8x + 21) \, dx = 16 + 42 = 58$

Thus, $\int_0^2 \sin(-3y)(8x + 21) \, dx = 58 \sin(-3y)$

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Step 4: Evaluate the outer integral with respect to $y$

$\int_0^2 58 \sin(-3y) \, dy = 58 \int_0^2 \sin(-3y) \, dy$

The integral of $\sin(-3y)$ is: $\int \sin(-3y) \, dy = \frac{-\cos(-3y)}{-3} = -\frac{1}{3} \cos(-3y)$

Evaluate from $y = 0$ to $y = 2$: $\int_0^2 \sin(-3y) \, dy = -\frac{1}{3} \left[ \cos(-3y) \right]_0^2 = -\frac{1}{3} \left( \cos(-6) - \cos(0) \right)$

$\cos(0) = 1, \quad \cos(-6) \text{ is a known value (can compute numerically if needed)}.$

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Step 5: Multiply and finalize

Combining the results: $\oint_C \mathbf{F} \cdot d\mathbf{r} = 58 \cdot -\frac{1}{3} \left( \cos(-6) - 1 \right)$

Numerical computation or exact trigonometric values would complete the solution. Let me know if you'd like the detailed computation finalized.