Question

Resolvendo a integral ydx + xdy ao longo da curva...

Answer 1

Temos uma integral de um campo vetorial $\overrightarrow{\mathbf{F}}(x,\,y)$ sobre uma curva parametrizada $R,$ sendo

$\overrightarrow{\mathbf{F}}(x,\,y)=(y,\,x)$

e a curva parametrizada

$R:~~\left\{ \!\begin{array}{l} x(t)=t\\\\ y(t)=t^2 \end{array} \right.~~~~~~~~0\le t\le 1.$

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Logo, a integral de linha é dada por

$\displaystyle\int_R\overrightarrow{\mathbf{F}}\cdot d\overrightarrow{\mathbf{r}}\\\\\\ =\int_R y\,dx+x\,dy\\\\\\ =\int_0^1 [y(t)\,x'(t)+x(t)\,y'(t)]\,dt\\\\\\ =\int_0^1 [t^2\,(t')+t\,(t^2)']\,dt\\\\\\ =\int_0^1 [t^2\cdot 1+t\cdot 2t]\,dt$

$=\displaystyle\int_0^1 [t^2+2t^2]\,dt\\\\\\ =\int_0^1 3t^2\,dt\\\\\\ =(t^3)|_0^1\\\\ =1^3-0^3\\\\ =1$


Resposta: alternativa $\text{e. }1.$