1. **Stating the problem:** We want to understand and apply Green's Theorem, which relates a line integral around a simple closed curve $C$ to a double integral over the plane region $D$ bounded by $C$. 2. **Green's Theorem formula:** $$\oint_C (P\,dx + Q\,dy) = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA$$ where $P$ and $Q$ are functions of $x$ and $y$ with continuous partial derivatives. 3. **Important rules:** - The curve $C$ must be positively oriented (counterclockwise). - The region $D$ must be simply connected (no holes). 4. **Explanation:** Green's Theorem converts a difficult line integral into a potentially easier double integral or vice versa. 5. **Example intermediate work:** Suppose $P = -y$ and $Q = x$, then $$\frac{\partial Q}{\partial x} = 1, \quad \frac{\partial P}{\partial y} = -1$$ So, $$\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 1 - (-1) = 2$$ 6. **Applying the theorem:** If $D$ is the unit circle $x^2 + y^2 \leq 1$, then $$\iint_D 2 \, dA = 2 \times \text{area}(D) = 2 \times \pi \times 1^2 = 2\pi$$ 7. **Final answer:** $$\oint_C (-y \, dx + x \, dy) = 2\pi$$ This shows how Green's Theorem relates the circulation around the unit circle to the area inside it multiplied by 2.