1. The Green's Theorem states that for a positively oriented, piecewise-smooth, simple closed curve $C$ in the plane and a region $D$ bounded by $C$, if $P(x,y)$ and $Q(x,y)$ have continuous partial derivatives on an open region containing $D$, then: $$ \oint_C (P\,dx + Q\,dy) = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA $$ 2. To prove this, we start by assuming $D$ is a type I region (vertical simple region), described by $a \leq x \leq b$ and $g_1(x) \leq y \leq g_2(x)$. 3. The curve $C$ is composed of four parts: the graphs of $y=g_1(x)$ from $x=a$ to $b$, $y=g_2(x)$ back from $b$ to $a$, and two vertical line segments connecting these curves at $x=a$ and $x=b$. 4. Consider the line integral over $C$: $$ \oint_C (P\,dx + Q\,dy) = \int_C P\,dx + \int_C Q\,dy $$ 5. Evaluate $\int_C P\,dx$ by parameterizing the top and bottom curves and the vertical segments. For vertical segments, $dx=0$ so the integral over these segments vanish. 6. Hence, $$ \int_C P\,dx = \int_a^b P(x, g_2(x)) \, dx - \int_a^b P(x, g_1(x)) \, dx = \int_a^b [P(x, g_2(x)) - P(x, g_1(x))] \, dx $$ 7. Similarly, evaluate $\int_C Q \, dy$. Using $dy = g_i'(x) \, dx$ for the curves, and for vertical segments $dy$ is vertical (integrals cancel). $$ \int_C Q \, dy = \int_a^b Q(x, g_2(x)) g_2'(x) \, dx - \int_a^b Q(x, g_1(x)) g_1'(x) \, dx $$ 8. Consider the double integral on the right-hand side: $$ \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA = \int_a^b \int_{g_1(x)}^{g_2(x)} \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dy \, dx $$ 9. By Fubini's theorem and integrating terms: $$ \int_a^b \left[ \int_{g_1(x)}^{g_2(x)} \frac{\partial Q}{\partial x} dy - \int_{g_1(x)}^{g_2(x)} \frac{\partial P}{\partial y} dy \right] \, dx $$ 10. Interchange integration and partial differentiation where allowed: $$ \int_a^b \left[ \frac{d}{dx} \int_{g_1(x)}^{g_2(x)} Q(x,y) dy - (P(x, g_2(x)) - P(x, g_1(x))) \right] dx $$ 11. Thus, $$ \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA = \int_a^b \frac{d}{dx} \int_{g_1(x)}^{g_2(x)} Q(x,y) dy \, dx - \int_a^b [P(x, g_2(x)) - P(x, g_1(x))] dx $$ 12. The first integral simplifies to: $$ \int_a^b \frac{d}{dx} \int_{g_1(x)}^{g_2(x)} Q(x,y) dy \, dx = \int_{g_1(b)}^{g_2(b)} Q(b,y) dy - \int_{g_1(a)}^{g_2(a)} Q(a,y) dy $$ 13. These terms cancel with the $\int_C Q \, dy$ terms along vertical segments, completing the equality: $$ \oint_C (P\, dx + Q \, dy) = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA $$ This completes the proof of Green's Theorem for a type I region. It can be extended to more general regions by decomposing them into such simple regions.