To verify Green's Theorem for the given vector field $\mathbf{F} = xy\mathbf{i} + (x^2 + y^2)\mathbf{j}$ and the curve $C$ defined by the arc of the parabola $y = x^2 - 4$ from $A(2, 0)$ to $B(4, 12)$, we follow these steps:

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Step 1: Green's Theorem Statement

Green's Theorem states: $\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_R \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dA$ where:

• $\mathbf{F} = P\mathbf{i} + Q\mathbf{j}$, so $P(x, y) = xy$ and $Q(x, y) = x^2 + y^2$.
• $C$ is a positively oriented, simple closed curve.
• $R$ is the region enclosed by $C$.

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Step 2: Calculate $\frac{\partial Q}{\partial x}$ and $\frac{\partial P}{\partial y}$

• $\frac{\partial Q}{\partial x} = \frac{\partial}{\partial x}(x^2 + y^2) = 2x$,
• $\frac{\partial P}{\partial y} = \frac{\partial}{\partial y}(xy) = x$.

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

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Step 3: Set up the region $R$

The parabola $y = x^2 - 4$ defines the curve $C$. The region $R$ lies below this curve and is bounded by $x \in [2, 4]$.

For the area integral, we will express $y$ as a function of $x$, which is given by $y = x^2 - 4$.

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Step 4: Compute the Double Integral

The double integral over $R$ becomes: $\iint_R x \, dA = \int_{x=2}^{x=4} \int_{y=-4}^{y=x^2-4} x \, dy \, dx.$

Evaluate the inner integral (over $y$): $\int_{y=-4}^{y=x^2-4} x \, dy = x \int_{y=-4}^{y=x^2-4} 1 \, dy = x \left[ y \right]_{-4}^{x^2-4} = x \left((x^2-4) - (-4)\right) = x(x^2 - 4 + 4) = x^3.$

Substitute into the outer integral (over $x$): $\int_{x=2}^{x=4} x^3 \, dx = \left[\frac{x^4}{4}\right]_{2}^{4}.$

Evaluate: $\left[\frac{x^4}{4}\right]_{2}^{4} = \frac{4^4}{4} - \frac{2^4}{4} = \frac{256}{4} - \frac{16}{4} = 64 - 4 = 60.$

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Step 5: Verify Line Integral

Using Green’s theorem, we have shown that the double integral is equal to $60$. Therefore, the line integral of $\mathbf{F}$ over $C$ must also be $60$, confirming Green's Theorem.

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Do you want further details on the computation or the geometric interpretation of Green’s theorem?

Related questions:

1. How does Green’s Theorem simplify evaluating a line integral?
2. Can Green’s Theorem be applied to regions with holes? Explain.
3. How do you parameterize a curve for line integral evaluation without Green’s theorem?
4. What are the conditions under which Green’s theorem is valid?
5. How do you verify a vector field is conservative using Green’s theorem?

Tip: Always check if the curve encloses a region in the positive (counterclockwise) orientation before applying Green’s Theorem.