Question

Teorema de Green verifique as hipóteses e calcule

Xy^2dx - x^2ydy onde c e o arco da parábola y=x^2 de (-1;1) a (1;1) seguido do segmento de reta (1;1) to ( -1;1)

Answer 1

$C$ é um curva fechada orientada positivamente (sentido anti-horário)

As componentes de $F(x,y)=(P,Q)=(xy^{2},-x^{2}y)$ possuem derivadas parciais contínuas

Portanto, o Teorema de Green é aplicável

Com isso, temos que

$\displaystyle\int_{C}xy^{2}\,dx-x^{2}y\,dy=\int_{C}P\,dx+Q\,dy=\iint_{D}\bigg[\dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y}\bigg]\,dx\,dy$

Achando as derivadas parciais de P e Q:

$\dfrac{\partial Q}{\partial x}=\dfrac{\partial}{\partial x}(-x^{2}y)=-2xy\\\\\\\dfrac{\partial P}{\partial y}=\dfrac{\partial}{\partial y}(xy^{2})=2xy\\\\\\\Rightarrow~~~\boxed{\boxed{\dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y}=-4xy}}$

Logo:

$\displaystyle\int_{C}F\cdot dC=\iint_{D}-4xy\,dx\,dy=-4\iint_{D}xy\,dx\,dy$

Onde $D$ é a região delimitada por $C$

Utilizaremos o Teorema de Fubini, pensando em x variando em extremos fixos (de -1 a 1) e, para cada x fixado, y varia de $y=x^{2}$ a $y=1$

$\displaystyle\int_{C}F\cdot dC=-4\int_{-1}^{1}\bigg[\int_{x^{2}}^{1}xy\,dy\bigg]dx=-4\int_{-1}^{1}\bigg[\dfrac{xy^{2}}{2}\bigg]_{y=x^{2}}^{y=1}dx\\\\\\=-\dfrac{4}{2}\int_{-1}^{1}(x-x^{3})dx=-2\bigg[\dfrac{~x^{2}}{2}-\dfrac{x^{4}}{4}\bigg]_{-1}^{1}\\\\\\=-2\bigg[\dfrac{1}{2}-\dfrac{1}{4}-\bigg(\dfrac{1}{2}-\dfrac{1}{4}\bigg)\bigg]=-2\cdot0\\\\\\\boxed{\boxed{\int_{C}xy^{2}\,dx-x^{2}y\,dy=0}}$