Teorema de Green - Exemplos Interativos

\[ \text{Área} = \oint_C -\frac{y}{2}dx + \frac{x}{2}dy \]

\[ \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = \frac{1}{2} - (-\frac{1}{2}) = 1 \]

\[ \begin{aligned} \text{Área} &= \frac{1}{2} \oint_C (-y dx + x dy) \\ &= \frac{1}{2} \int_0^{2\pi} [-b\sin t(-a\sin t) + a\cos t(b\cos t)] dt \\ &= \frac{ab}{2} \int_0^{2\pi} (\sin^2 t + \cos^2 t) dt \\ &= \frac{ab}{2} \int_0^{2\pi} 1 \, dt = \pi ab \end{aligned} \]

\[ \begin{aligned} \oint_C P\,dx + Q\,dy &= \int_0^{2\pi} [-\sin^3 t (-\sin t) + \cos^3 t (\cos t)] dt \\ &= \int_0^{2\pi} (\sin^4 t + \cos^4 t) dt \\ &= \int_0^{2\pi} \left(1 - \frac{1}{2}\sin^2 2t\right) dt \quad \text{(usando identidades trig.)} \\ &= 2\pi - \frac{1}{2} \cdot \frac{2\pi}{2} = \frac{3\pi}{2} \end{aligned} \]

\[ \frac{\partial Q}{\partial x} = 3x^2, \quad \frac{\partial P}{\partial y} = -3y^2 \] \[ \Rightarrow \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 3(x^2 + y^2) \]

\[ \iint_R 3(x^2 + y^2) dA = 3 \int_0^{2\pi} \int_0^1 r^2 \cdot r \, dr \, d\theta = 3 \cdot 2\pi \cdot \frac{1}{4} = \frac{3\pi}{2} \]