I have to following exercise (with solutions): 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 $(1-t^2,t^2)$?
Building line integrals
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calculus
integration
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0$F$ is not $P$, $F$ is $(P,Q)$. They are integrating $F(r(t))\cdot r'(t)$. – 2012-12-12
1 Answers
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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.