Evalúe la integral de línea sobre la curva C

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Hola Johanna!

F(x,y)=P(x,y)i+Q(x,y)j=(sent, -2cost)

C(t)=(sent,-2cost)

C'(t)=(cost, 2 sent)

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$$\begin{align}&\int_C \ F·dR= \int_C P(x,y) dx+Q(x,y)dy=\\&\\&\int_C2xy\ dx\ + \ (x-2y)dy=\\&\\&\int_0^{\pi}2sent(-2cost)cost\ \ +(sent+4cost)2 sent\ \ \ \ dt=\\&\\&\int_0^{\pi}-4cos^2t\ sent\ +2\ sen^2t\ +\ 8\ sent\ \cos t\ \ \ \ dt=\\&\\&4 \frac{\cos^3t} 3\ +\ 2\ \frac 1 2 [t- \frac 1 2 sen(2t)]\ \ +\ 4\ sen^2 t  \Bigg|_0^{\pi}=\\&\\&\frac 4 3 (-1-1)\ \ +\ [ \pi-sen(2 \pi)-(0-sen0)] \ +\ 4 sen^2 \pi- 4sen^20=\\&\\&- \frac 8 3+ \pi\\&\\&\end{align}$$

Saludos

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