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I have to following exercise (with solutions): Problem But I don't understand how they build the integrals from $\int_{\partial D} P dx + Q dy$, as it should be $F(r(t))$, with $F=P$ and $r(t)$ according to the parameterization. How do they get $(1-t^2,t^2)$?

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$F$ is not $P$, $F$ is $(P,Q)$. They are integrating $F(r(t))\cdot r'(t)$. –  Sigur Dec 12 '12 at 0:51
    
@Sigur thanks, now it makes sense. Post it as answer so I can accept it. –  Tass Dec 12 '12 at 0:53

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The method is to integrate the field $F=(P,Q)$ on the curve $r(t)$. So you need to compute $$\int F(r(t))\cdot r'(t)dt.$$ Note that you have two curves so you have to sum both integrals.

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