Question

5) Calcule a integral pelo teorema de Green ∫c y² dx + 3xy dy.
Sendo a curva representada no gráfico
da = r dr dθ
0 ≤θ ≤ π
1≤ r ≤ 2

x = r cos θ
y = r sen θ

Veja anexo:

Answer 1

$\\ \begin{enumerate} \\ \\ De maneira analoga, \\ ja que temos duas componentes "x" e "y" do campo: \\ \\ Rot(F) = ( dFy/dx - dFx/dy )k \\ \\ dFy/dx = 3y \\ \\ dFx/dy = 2y \\ \\ \therefore \\ \\ Rot(F) = (3y - 2y)k = yk \\ \\ Logo, integral fica: \\ \\ = \displaystyle \int\limits^{ \pi }_0 {} \displaystyle \int\limits^{2 }_1 {} (y) \cdot rdrd \beta \\ \\ Com, y = rsen \beta$

$\\ = \displaystyle \int\limits^{ \pi }_0 {} \displaystyle \int\limits^{2 }_1 {} (rsen \beta ) \cdot rdrd \beta \\ \\ = \displaystyle \int\limits^{ \pi }_0 {} \displaystyle \int\limits^{2 }_1 {} (sen \beta ) \cdot r^2drd \beta \\ \\ = \displaystyle \int\limits^{\pi }_0 sen \beta \cdot \frac{r^3}{3} |_1^2 d \beta \\ \\ = \displaystyle \int\limits^{ \pi }_0 sen \beta \cdot (\frac{2^3 -1^3}{3} )d \beta$

$\\ = \frac{7}{3} \cdot \displaystyle \int\limits^{ \pi }_0 sen \beta \cdot d \beta \\ \\ = \frac{7}{3} \cdot (-cos \beta ) |(0, \pi ) \\ \\ = \frac{7}{3} \cdot (1 - (-1)) \\ \\= \frac{14}{3}$