Evaluate $\int_C (2x^3-y^3)dx+(x^3+y^3)dy$, where $C$ is the unit circle.
Answer
By Green's Theorem we have \[\int_C(2x^3-y^3)dx+(x^3+y^3)dy=\iint_{D} \frac{\partial (x^3+y^3)}{\partial x}-\frac{\partial (2x^3-y^3)}{\partial y}dxdy\] \[ =\iint_{D} 3x^2+3y^2 dx dy=\int_{0}^{2 \pi} \int_0^{1}3r^2 r dr d\theta=2\pi \int_0^{1}3r^3 \] \[=2\pi(\frac{3}{4})=\frac{3\pi}{2}.\]
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