Chapter 9: Vector Calculus

Section 9.10: Green's Theorem

Example 9.10.2

Use Green's theorem to show that the area inside the plane region $R$ is given by $-{∳}_{C}y \mathrm{dx}$. Solution Take $\mathbf{F}\=y\mathbf{ }\mathbf{j}$ in the divergence-form of Green's theorem. Since $f\=0$ and $g\=x$, $\int {\int }_R\left(f_x\+g_y\right) \mathrm{dA}$ = ${∳}_{C}f \mathrm{dy}-g \mathrm{dx}$ becomes $\int {\int }_{R}1 \mathrm{dA}$ = $A$ = ${∳}_C\left(-y\right) \mathrm{dx}$.

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