Compute $\oint_C y^2dx+3xydy $ where where $C$ is the counter clickwise oriented boundary of upper-half unit disk
Answer
Let $D$ be the upper-half unit disk. Using Green's theorem we get
\[\oint_C y^2dx+3xydy =\iint_D \frac{\partial}{\partial x}(3xy)-\frac{\partial}{\partial y}(y^2)dxdy\]
\[=\iint _D y dx dy=\int_{-1}^{1}\int_{0}^{\sqrt{1-x^2}}ydydx\]
\[=\int_{-1}^{1} \frac{y^2}{2}\Big|_0^{\sqrt{1-x^2}}dx =\int_{-1}^{1} \frac{1-x^2}{2}\]
\[=\frac{1}{2}(x-\frac{x^3}{3})\Big|_{-1}^{1}=\frac{1}{2}\Big(\frac{2}{3}- (-\frac{2}{3})\Big)=\frac{2}{3}.\]
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